K. Galsgaard, V. S. Titov and T. Neukirch- Magnetic Pinching of Hyperbolic Flux Tubes. II. Dynamic Numerical Model

8/3/2019 K. Galsgaard, V. S. Titov and T. Neukirch- Magnetic Pinching of Hyperbolic Flux Tubes. II. Dynamic Numerical Model

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MAGNETIC PINCHING OF HYPERBOLIC FLUX TUBES. II. DYNAMIC NUMERICAL MODELK. Galsgaard 1

NielsBohr Institutefor Astronomy, Physics, and Geophysics, JulieMaries vej 30, 2300Copenhagen , Denmark;[email protected]

V. S. Titov

TheoretischePhysik IV, Ruhr-Universitat Bochum, 44780 Bochum, Germany

and

T. Neukirch

School of Mathematics andStatistics, University of Saint Andrews,Saint Andrews, KY16 9SS,ScotlandReceived 2003 April 1; accepted 2003 June 2

ABSTRACTIn this paper we present the results of a series of numerical experiments that extend and supplement the

recent analytical investigations by Titov et al. of the formation of strong current layers in coronal magneticelds containing hyperbolic ux tubes (HFTs). The term hyperbolic refers to the special geometricalproperties of the magnetic eld, whereas the topology of the eld is simple; i.e., there are no magnetic nullpoints and separatrix lines or surfaces associated with them inside the coronal volume. However, the eldlines passing through a hyperbolic ux tube show a large variation in the mapping between their photosphericendpoints. On the basis of analytical estimates, it has been suggested by Titov et al. that HFTs are preferred

locations for the formation of strong current layers in coronal magnetic elds with trivial topologies,provided the driving motions on the photospheric boundary are of a special type. Such motions must haveshearing components that are applied across narrow HFT feet as if trying to twist it. This system of motionsis then capable of causing a pinching deformation of the HFT by a sustained stagnation point ow inside theHFT. The numerical experiments presented in this paper clearly conrm this suggestion. HFTs are genericfeatures of geometrically complex but topologically trivial magnetic elds, and therefore our results are veryimportant for understanding magnetic reconnection in such elds, since reconnection is occurringpreferentially at locations with strong current densities.Subject headings: Sun: ares Sun: magnetic elds

1. INTRODUCTION

An important aspect of the dynamical evolution of a mag-

netized plasma is the process of magnetic reconnection.Magnetic eld penetrates most of the universe, and in manyregions the magnetic forces are small compared with theplasma forces. Inmost of space the magnetic eld is frozen tothe plasma, and it is advected around by the generallyturbulent plasma motions. This tangles and amplies themagnetic eld, and if the magnetic eld were not allowed tochange its connectivity, magnetic tension forces wouldamplify to a level at which they could suppress the turbulentplasma motions. This is not the case in general;magnetic eldlines are allowed to diffuse through the plasma and changetheir connectivity when large enough electric current concen-trations are reached locally. To know how and where mag-netic reconnection proceeds is important for understanding

the dynamical evolution of a stressed magnetic eld. In threedimensions this is a complicated issue, with several possiblemagnetic topologies being able to undergo magnetic recon-nection. Extending our knowledge from two-dimensionalmagnetic reconnection makes three-dimensional magneticnull points likely spatial locations for driving reconnection(Lau & Finn 1990; Craig et al. 1995; Priest & Titov 1996;Rickard & Titov 1996; Galsgaard, Rickard, & Reddy 1997).Apart from reconnection at null points and their connec-tions, it is known that magnetic reconnection may also take

place at special locations in a magnetic eld without a mag-netic singularity (Hesse & Schindler 1988; Otto 1995; Priest &Demoulin 1995; Galsgaard & Nordlund 1996; Inverarity& Titov 1997; Hornig & Rastatter 1997; Titov, Demoulin, &Hornig1999).

Priest & Demoulin (1995) used an ideal kinematicapproach to show that certain imposed boundary motionscan generate large eld line velocities at locations where theeld line mapping between two boundaries of the consid-ered volume changes particularly rapidly. On the basis of the examples presented in their paper, Priest & De moulin(1995) concluded that these locations have a layer-likespatial structure similar to separatrices, and therefore theycalled them quasi-separatrix layers (QSLs). Under idealconditions the eld line velocity is identical to the plasmavelocity perpendicular to the magnetic eld. Priest &Demoulin (1995) suggested that under the conditions dis-cussed by them, the eld line velocity would eventuallybecome larger than the local Alfve n velocity. As the plasmavelocity cannot exceed the local Alfven velocity this impliesthat plasma velocity and eld line velocity must decouplein this case, which in turn implies the existence of some non-ideal process, viz., reconnection. Therefore, Priest &Demoulin (1995) suggested that QSLs are genericallyfavorable places for magnetic reconnection.

The limitation of their kinematic analysis is that it doesnot take into account the dynamical changes of the mag-netic eld and the corresponding changes of the eld linestructure. To investigate this Galsgaard (2000) made a seriesof numerical experiments using the full set of MHD

1 Permanent address: School of Mathematics and Statistics, Universityof Saint Andrews,Saint Andrews,KY16 9SS,Scotland.

The Astrophysical Journal , 595:506516, 2003September 20# 2003.The American Astronomical Society.All rights reserved. Printedin U.S.A.

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equations, and these showed that the presence of a QSLalone is not a sufficient condition for reconnection. It hasalso recently been shown that the rst denition of a QSL isnot invariant to the direction of the eld line mapping(Titov et al. 1999) and that they are geometrical objectsrather than topological ones that can be removed by suit-able smooth deformations of the magnetic conguration(Titov & Hornig 2002).

In a quadrupolar magnetic eld conguration formed bytwo bipolar regions, the separation of ux on the photo-spheric plane can be understood by the intersection of twoQSLs that in the limit of point sources converges to two fansurfaces (or separatrix surfaces) intersecting along a separa-tor eld line. This conguration is known as a hyperbolicux tube (HFT; Titov et al. 2002). It is known that the sepa-rator line is a favorable site for current accumulation whenthe system is perturbed (Sweet 1969; Gorbachev & Somov1988; Lau & Finn 1990; Galsgaard, Priest, & Nordlund2000; Longcope 2001). It can therefore be expected that thesame region in a eld dened by distributed but well-pronounced sources without nulls in the domain will also bea preferred location for current accumulation.

The present paper is the second in a series of three papers(Titov, Galsgaard, & Neukirch 2003, hereafter Paper I) inwhich we intend to clarify the physical conditions underwhich strong current layers form in HFTs. In all papers weinvestigate a particular type of magnetic eld that containsan HFT. This magnetic eld is subjected to differentboundary perturbations inherent to the generic type of photospheric motions, and the response of the HFT to theseperturbations is studied. Paper I does this using ananalytical approach, which shows that exponential growthof the current can be reached when the structure is stressedin a way in which a stagnation ow is initiated around theseparator-like line by the perturbation of the magnetic eld.This paper adopts the full set of MHD equations and solvesthem numerically. In Paper III (T. Neukirch, K. Galsgaard,& V. S. Titov 2003, in preparation) we will use a Lagrangianrelaxation code to investigate the nonlinear force-free solu-tions of the quasi-static evolution of an HFT and thecurrent buildup due to slow changes in the boundary condi-tions. Together these papers intend to show that the forma-tion of strong current layers in HFTs (pinching) is thegeneric process and therefore very important for our under-standing of reconnection in general and of solar arephenomena in particular.

The layout of the paper is as follows. In x 2 we introducethe initial conditions and the perturbations. The numericalapproach is described in x 3. Section 4 discusses the experi-ments and their implications for our understanding of current accumulation in HFTs. Finally, in x 5, the resultsare summarized and conclusions are drawn.

2. EXPERIMENT SETUP

Our aim is to test the hypothesis that inside an HFT astrong current density only develops if the HFT is stressedby a specic type of boundary motions. The testing is doneby performing a set of numerical experiments. For theexperiments we construct a magnetic eld containing anHFT by following the guidelines provided in Paper I. Thismagnetic eld is a potential eld that is periodic in the x-and y-directions and nonperiodic in the z-direction. Thenumerical box extends over one wavelength of periodicity in

the x- and y-directions and is bounded by two boundaries atz 0 and 1 (in suitably normalized units) in the z-direction.On the two z-boundaries we impose the z-component of themagnetic eld. On each boundary we combine two Gaus-sian ux concentrations (normalized to peak magnitude 1)with a weak constant background magnetic eld of magni-tude 0.1 (Fig. 1). The lines connecting the two Gaussian uxconcentrations on the top ( z 1) and bottom ( z 0)boundaries have an angle of 90 with respect to each other.The periodic potential eld corresponding to these bound-ary conditions has an HFT in the center of the domain inthe periodic direction. It also shows the characteristic strongvariation in the eld line mapping from one footpoint to theother. Halfway between the two boundaries the potentialmagnetic eld is nearly uniform with an average eldstrength of 0.16. Due to the periodic boundary conditionsthe HFT conguration is repeated innitely often in the x-and y-directions. This facilitates the numerical setupenormously, but it also introduces additional complexityinto the eld line connectivity compared with the non-periodic eld studied in Paper I (Fig. 2; cf. also Fig. 2 of Paper I). This additional complexity causes the appearanceof secondary locations of current accumulation inside thenumerical domain.

This magnetic eld contains a number of regions wherethe eld line mapping between the two boundaries has alarge gradient. A more detailed explanation of this for thenonperiodic case is given in Paper I. The structure of themagnetic eld line mapping is shown in Figure 3, where eldlines are traced from a ring around the center of one uxsource from each of the two z-boundaries. Notice how theux from the source on one boundary mostly connects toweak eld regions on the opposite boundary and especiallyhow the eld lines around the central separator line divergefrom this region. Strong currents would therefore naively beexpected to accumulate here as the system is stressed.

Fig. 1. Flux pattern on the driving boundaries. The pattern in the twooppositeboundaries are rotated 90 relative to one another.

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To investigate how HFTs are accumulating electric cur-rent as soon as the magnetic eld around them is stressed,we have set up a number of experiments. In these experi-ments, the ux patterns on the two z-boundaries areadvected in different directions using a sinusoidal shear motion and with different driving amplitudes. The imposeddriving prole has a wavelength equal to the extent of thenumerical domain and is, like the ux concentrations,rotated by 90 between the two boundaries. The shearmotion is imposed such that the sources are advected in adirection perpendicular to the line initially connecting thesources. Figure 4 shows the structure of the driver relativeto the location of the two sources. The experiments are car-ried out by changing the sign of the amplitude of the driving

velocity on the bottom boundary and by changing theamplitude of the driving.

For the initial conditions the plasma is assumed to beat rest, and the plasma is less than unity away fromthe sources. In the units of the code, described below,this is achieved by setting the thermal energy and densityto 0.1. This provides an Alfven velocity in the centralpart of the domain of 0.49, while the uniform soundspeed becomes 0.33. This setup gives an Alfven crossingtime of about 2 time units. In the same units the maxi-mum driving velocity is 0.1 for the standard experiment,which is about 32% of the Alfven velocity at the back-ground eld at the source planes and 2.9% of the peakAlfven velocity of the ux concentrations.

Fig. 2. Structure of the potential magnetic eld generated by two grids of ctive point sources of opposite signs. The grids are periodic in the x-, y-directions and located at a given distance outside the boundary planes z 0 and 1 so that each periodic cell contains two positive and two negative sources.Right : Field linesforming the skeleton of the studied HFT. Left : Fictive nullpoints and separatrix lines belonging to the front boundaryof the cell. Becauseof the symmetry of the grids there are similar nulls and separatrices at the back boundary. At the lateral boundaries, they are also present and obtained fromthe shown ones by rotating them 90 around the center of the cell with subsequent ipping of the nulls and eld line arrows about the plane z 0:5. Thesources are represented by small dark gray circles; the corresponding ux concentrations on the top and bottom boundaries of the right cell are represented bylarge solid ( positive ) anddashed ( negative ) circles, while the fan separatrix planes at the nulls are represented by large lightgray circles.

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3. NUMERICAL APPROACH

To follow the dynamical evolution of the magnetizedplasma as the ux sources on the two boundaries areadvected in time, we solve the full set of MHD equations in

a three-dimensional Cartesian domain,@@ t

Dx u; 1

@ u@ t

Dx uu DP J B ; 2

@ e@ t

Dx eu P Dx u Q Joule Q visc ; 3

@ B @ t

D E ; 4

E u B J ; 5J D B ; 6

with density , velocity u, thermal energy e, magneticeld B , electric eld E , magnetic resistivity , electriccurrent density J , viscous stress tensor , gas pressureP e 1, viscous dissipation Qvisc, and Jouledissipation Q Joule , respectively. An ideal gas with 53is assumed. The equations are nondimensionalized by set-ting the magnetic permeability at l 0 1 and the gas con-stant at R0 l (the mean molecular weight). One time unitis equivalent to the Alfve n transit time of a unit length atwhich both B j j and are set to 1.

The equations are solved using staggered grids. A sixth-order method is applied to derive the partial derivatives,and a fth-order method is used for doing interpolation.Viscosity and magnetic resistivity are both handled using acombined second- and fourth-order method with a discon-tinuous shock-capturing mechanism to provide the highestpossible spatial resolution for the given numerical resolu-tion. The solution is advanced in time using a third-orderpredictor-corrector method. 2

4. THE EXPERIMENTS

The driving velocity on the two boundaries is slowlyincreased from zero at the start of the experiment toward itsmaximum. When this value is reached, the amplitude ismaintained constant for the rest of the experiment. Theimpact on the magnetic eld is to launch an Alfve n wavefrom the boundaries that propagates into the domain. Thiswave creates a weak current concentration at its front. As

Fig. 3. Field line structure of the initial potential magnetic eld. Fieldlines are traced from circles around the centers of two of the ux sources.The images show two different orientations of the three-dimensionaldomain.

Fig. 4. Structure of the driver relative the locations of the two uxsources. The full line represents the starting points of tracer particlesadvected with the driving ow. The dashed line shows their position at alater time. The concentric circles show the initial locations of the twosources.

2 A basic description of the code is available in Nordlund & Galsgaard(1997) and at http://www.astro.ku.dk/kg.

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driving is imposed on both boundaries the simple wavefronts soon become complicated as they interact; they arereected from the boundaries and propagate with differentspeeds depending on their spatial location. As the uxsources are systematically advected all the time, the orienta-tion of the eld lines connecting the two boundaries slowlychanges. This systematic stretching and bending of the mag-netic eld lines generates different direct current systemsthat depend on the imposed driving prole. The issue here isto understand where in the magnetic eld conguration cur-rents tend to concentrate and why the accumulation takesplace at these particular locations.

Six experiments are discussed here, in which the differen-ces are in the sign of the driving amplitude on the bottomboundary, the driving amplitude itself, and the viscosity.Table 1 summarizes the main parameters of the experi-ments, showing the naming of the experiments, the drivingamplitude, the sign and scale value of the amplitude on thebottom boundary, relative viscosity, and the time of termi-nation of the experiment. The negative/positive value of sign in Table 1 refers to the combined effect of the drivingpattern on the two z-boundaries. A negative value implies a turning motion of the central HFT structure, while apositive value represents a twisting motion of the HFTstructure. The implications of these boundary ows areillustrated in Figure 5. The result from Paper I is that onlythe twisting motion of the HFT generates a fast growth of current along the HFT. By relative viscosity it is impliedthat the coefficients controlling the nonlinear viscosity inthe code are increased relative to the minimum values usedin ve of the experiments. In the turn experiment, twoequally strong current structures are generated away fromthe main HFT located at the center of the domain, while inthe twist_13 cases only one strong current concentra-tion forms in the center of the domain initially, followed bysecondary current concentration later in the experiment.This is in full agreement with the results in Paper I. The shear experiment shows a completely different currentaccumulation. As the driving in this case is a linear combi-nation (the mean value) of turn and twist_1, in whichonly the footpoints at the top boundary are advected, a sim-ple shear of the HFT is provided. Paper I suggests that thisshould generate a current structure that has features of boththe turn and the twist_1 experiments. This is furtherconrmed by twist_4, which is a different linear combina-tion of turn and twist_1 (0.4 times turn and 0.6times twist_1 ) and produces a weaker twist of the HFTthan twist_1. In shear no signicant current accumulates,while in twist_4 current grows in the central region, as in twist_1, but with a much slower rate.

Figure 6 shows the current distribution of the z-current inthe x ; y-plane halfway between the two driving boundaries

(top ) at time 7 (shear 0.7), where the shear is dened asV drive t and at the end ( bottom ) of the turn, twist_1,and shear experiments. From the panels it is seen howthe current density concentrates in three signicantly differ-ent patterns depending on the sign and amplitude of thedriver on the lower boundary at the two different times of the experiments. At time 7 (shear 0.7) it is found that twosymmetric current concentrations are formed away fromthe center of the x ; y-plane in turn, while a strong cur-rent concentration is formed at the center of the plane in twist_1. In shear only a slightly enhanced current con-centration is formed at the center of the x ; y-plane, while alarge-scale current structure is generated as a consequenceof the simple shear motion imposed by the driving on oneboundary only. Toward the end of twist_1, the currentstructure has changed, and the initially secondary currentconcentrations that are active in turn are as strong as thecurrent layer in the central region of the domain. Thischange in location of the current accumulations occurs fortwo reasons. First, the magnetic structure dening the cen-tral current layer starts to undergo a magnetic reconnectionthat diffuses the current faster than it can be built up. Sec-ond, the continued driving eventually starts to stress the sec-ondary regions of strong divergence in the magnetic eld

Fig. 5. Effect of the turn and twist driving on the structure of theHFT.

TABLE 1General Data for the Experiments

Experiment V drive Sign Viscosity EndTime

turn..................... 0.1 1 1 14.1 twist_1 ................ 0.1 1 1 13.7 twist_2 ................ 0.025 1 1 31.7 twist_3 ................ 0.025 1 10 14.6 twist_4 ................ 0.1 0.2 1 12.2 shear ................... 0.1 0 1 12.1



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line mapping, and current starts to build up at these loca-tionsa clear effect of the imposed periodicity in the x; y-plane.

The extended loop structures in the current concentra-tions that are found in the later phases of twist_1 areclear indications of a pileup of the current generated by thereconnection jets originating from the central current sheet.

In Figure 7 the time-dependent increase of the maximumand minimum z-current are shown for ve experiments as afunction of time. Experiment twist_3 is not shown, as itscurve is nearly identical to that of twist_2. The bumps inthe curves are due to the wave fronts interacting with eachother as they bounce between the boundaries. Here it isnoticed that the peak currents in turn have the same ampli-tude but with opposite signs. Comparing the twist_12shows that with decreasing the driving velocity by a factorof 4 the current strength is decreased by nearly the same fac-tor. This implies that there is a linear relation between thetransport distance of the magnetic footpoints and the mag-nitude of the current density at early stages of the evolution,when the driving velocity is below the Alfven velocity in thecenter of the domain (see next paragraph). It is noticed thatat later times the z-component of the current density growsfaster fora slower driving than fora faster onecharacterizedby the same resulting shear distance. This is expected, since

Fig. 6. Distribution of the current component along the z-direction in the x; y-plane halfway between the two driving boundaries for turn, twist_1,and shear. The top panels represent t 7 (shear 0.7), and the bottom panels show the current distribution at the end of the experiments. The main panelshows a shaded surface plot, while the small image at the bottom left of each frame shows the current structure as a scaled image with black being negativeandwhite positive.Each image is scaled to the minimummaximumrange in the individual datasets.

Fig. 7. Time-dependent positive/negative peak value of j z in the x; y-plane halfway between the two driving boundaries. Top full line is turn ;the dashed line and dot-dashed line are the positive and negative currentfrom twist_1, with the dot-dot-dashed line representing twist_2; thedotted line is twist_3; the bottom full line is shear, and thelong-dashedline represents twist_4.



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in the last case the magnetic eld has a longer time forresponding to the imposed boundary stress. In addition, thisindicates that the resistive effects are negligible for restrain-ing the current growth in early stages of the experiments.Comparing the evolution of the peak current density in twist_2 and twist_3 shows that it is not effected signif-icantly by the 10 times increase of the viscosity.

The parameters used in twist_1 provide a representa-tive HFT pinching, and it is therefore the only experimentwhose results will be discussed below in detail. In Paper Itwo predictions of the current growth in the center of theHFT halfway between the boundaries were given inequation (25) and equation (41) for the kinematic and,respectively, force-free approximations of the developmentof the perturbedmagnetic structure. For convenience they are repeated here:

j zkinematic %e2tth

l 01

B k2hL 7

j zforce - free %e2tth

l 01

B k2hL

1 e2tt

0:91hl shB k 0:57

l shL

2

" #: 8In Paper I, a hyperbolic tangent driving prole has beenused on both boundaries with l sh being the scale length of variation of this velocity prole and V drive being its ampli-tude. Furthermore, tt V drive t=2l sh, and h determinesthe strength of the two-dimensional magnetic x-point in thex ; y plane halfway between the two z-boundaries for theinitial magnetic eld (see eq. [3] in Paper I), and B k is the z-component in the same plane (see eq. [4] in Paper I). Thevarious coefficients in these expressions only depend on theinitial magnetic eld and the imposed driving prole. Thecurrent in the analytical solutions is found to grow exponen-

tial with tt. For small tt the exponential factor e2tt

is replacedby 2 sinh2tt. This correction is essential for the compari-son with our simulations, because the time tt is small com-pared with the driving time of the system. The current

growths for these two estimates are presented along with thepeak current densities in twist_1 and twist_2 in Fig-ure 8. It shows that the growth of these values in the numeri-cal experiments closely follows the kinematic prole until ashear distance of 0.7 is reached. After that the limitednumerical resolution comes into play by effectively enhanc-ing the resistivity and thereby restraining the further growthof current. This becomes especially evident for a shear dis-tance of 1.0, at which such a restraining becomes so strongthat the maximum of the current density in the spatialdistribution moves out of the center of the current sheet.

To make the above comparison of the peak-current evo-lutions, the appropriate expression of tt was derived. In ournumerical experiments a sinusoidal velocity prole was usedinstead. A suitable value of l sh was estimated from this pro-le, while the rest of parameters required for equations (7)and (8) were determined from the initial magnetic eld.Because of some ambiguity in the estimate of l sh the consid-ered theoretical curves of the current growth may noticeablyshift, so they mainly represent the trend rather than theexact behavior of the current values.

Why does the change in the direction of the driver on thebottom z-boundary cause the above differences in the cur-rent location and its strength? A good insight into this ques-tion is provided by the consideration of the eld linemapping between two boundaries depending on the value of shears imposed on these boundaries. Figure 9 shows thevar-iation of the eld line arrangement in turn, twist_1,and shear at t 8:1, corresponding to the shear of 0.81.The left panels in this gure refer to turn, and they dem-onstrate that the corresponding eld line structure is rathersimilar to the initial one shown in Figure 3. The ux tubesbecome noticeably squashed only close to the lateral boun-daries, where secondary spikes of current are developed(Fig. 6, left panels ). The middle panels in Figure 9 refer to twist_1, where a much stronger squashing of the uxtubes takes place when the eld lines approach the oppositeboundaries. The squashing of the considered ux tubesoccurs here in nearly parallel rather than perpendiculardirections, as it was in turn. Additionally, these squashedux tubes are clustered around the central axes, which is ingood agreement with the abovementioned formation of acurrent layer in the middle of the HFT. The right panels inFigure 9 refer to shear, in which only one of the consid-ered ux tubes is essentially squashed. The latter occurs atthe boundary, where the footpoints are advected to producea weak localized current accumulation. This situation isactually similar to the one considered earlier by Galsgaard(2000) for a more simple X-point conguration with a non-zero longitudinal magnetic eld.

Titov et al. (1999) showed that an invariant way of measuring the degree of squashing the magnetic ux tubesin a conguration is given by

Q @ x0=@ x 2 @ x0=@ y 2 @ y0=@ x 2 @ y0=@ y 2

@ x0=@ x @ y0=@ y @ x0=@ y @ y0=@ x j j: 9

The partial derivatives represent the four elements in theJacobian matrix describing the mapping of x; y positionsfrom one z-boundary to the other.

If an innitesimally small cross section of an elementalux tube does not experience a squashing when it is mappedalong the eld lines from one boundary to the other, thenthe corresponding Q takes the value of 2. In other cases

Fig. 8. Time-dependent increase of the shear current determined fromeq. (25)( gray ) and eq. (42)( black ) in Paper I. Thedot-dashedlinerepresentsthe peak current in twist_1, and the dotted line represents the peak cur-rent in twist_2.



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Q is larger than 2, and it grows monotonically with thesquashing of elemental ux tubes.

Using this denition, one can compute Q as a function of time for our six experiments. Here it has been achieved bytracing magnetic eld lines from a uniform grid on one z-boundary, containing 100 100 starting positions, to theother z-boundary. From the vicinity of these positions threeeld lines were traced providing the basis for estimating theelements of the Jacobian matrix locally. The straight-forward calculation of Q from these results is difficultbecause of the numerical truncation errors, which are toolarge for the determinant of Jacobi matrix entering into thedenominator of equation (9). However, since the magneticux is conserved in thevolume, thevalue of this determinantis equal to the ratio of the normal components of the mag-netic eld at the footpoints of a given eld line (see thedetails in Titov, Hornig, & De moulin 2002). With such asubstitution in equation (9), Q has been computed in each of the six experiments. It is of particular interest to study thevalue of Q for the elemental ux tube passing through thecenter of the HFT, because this region is most representa-tive for the effect we study and less than others inuenced bythe lateral periodic boundary conditions. Figure 10 repre-sents this value as a function of maximal shear distance at agiven moment for each of the experiments. In all of them Qgrows exponentially with time after an initial period andtends to grow more slowly for large shear distances. Thisbehavior of Q is in agreement with the results of Paper I, inparticular, with its equations (6) and equation (7) demon-strating the exponential dependence of Q on the averagegradient of transverse eld component in the initial HFT.One can expect that a similar exponential expression is also

valid for Q at the axis of our perturbed HFT, in which sucha eld gradient will grow with time under the imposed shearmotions at the HFT feet by giving the observed exponentialdependence of Q on shear. A somewhat faster growth of Qin twist_1 and twist_2 can be explained by the actionof stagnation ows in the middle of HFT, which causes anexponentially fast squashing of the corresponding Lagran-gian elements of plasma (see eq. [19] of Paper I for thedegree of such a squashing). The reduction in the growth of

Fig. 9. Field line connectivityin turn, twist_2, and shear areshown from left to right using eld line tracesstarted from a ring centered on thefoursources. Two different orientations are shown for the three experiments ( top and bottom rows ). The contour lines show the locations of the sources on the twoboundaries.

Fig. 10. Maximum value of the Q factor dened in eq. (9) is calculatedfor the six experiments using the ux ratio for the denominator. The sixexperiments have the following signatures: turn: long-dashed line; twist_1: full line; twist_2: short-dashed line; shear : triple-dot dashed line; and twist_4: dotted line.



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Q toward the end of the experiments indicates that magneticreconnection comes into play at this stage by restrainingfurther divergence of eld-line footpoints.

Thus, one can notice that the local growth of Q at the axisof theHFTis notalwaysaccompanied by the correspondinggrowth of current density. All the experiments start fromthe same structure of the magnetic eld with the largestvalue of Q at the center of the domain. The experiments dif-fer in the way the magnetic eld lines react to the differentchanges of the location of the footpoints. The clue to under-standing the physical reason of the differences in turnand twist_1 can be found by comparing the correspond-ing patterns of the ow and distribution of J z in the z-planehalfway between the two driving boundaries. Figure 11demonstrates that the location where the strong currentdensity builds up is exactly the location where the magneticforces form a stagnation ow. The direction and amplitudeof the current density then depends on the properties of themagnetic eld at this place. Galsgaard & Nordlund (1996)showed that a stagnation ow can initiate an exponentialgrowth of the current density with time if the magnetic eldat the stagnation point has an appropriate structure. Anexponential growth of the current density at the center of the forming current layer is also theoretically predicted inPaper I (see eq. [25]) for the quasi-static evolution of theconguration. In our dynamical simulations, however, thisis not the case, even though the same driver has been usedand the shear distances were comparable in both numericalexperiments. How can this be? In Galsgaard & Nordlund(1996) an initial uniform magnetic eld is braided by two

subsequent boundary shears, which are oppositely directedon the two boundaries and put one after another in perpen-dicular directions (Figs. 11 a 11c). The rst shear produces asheared magnetic conguration with a slablike geometry.The second shear, being perpendicular to the slab, forceseach of the two eld lines on the opposite sides of the slab tomove toward one another by strongly linking them up.Therefore, at large amplitudes of the shears a strong mag-netic tension in the corresponding eld lines is developed bypulling them toward each other and thereby forming astrong current layer in between. The fast growth of currentdensity stops only at late stages of the process because of theresistive effect.

In our present experiments the initial magnetic eld ishighly inhomogeneous and stressed by shearing motionsthat are imposed on the boundaries in perpendicular direc-tions. The initial eld lines belonging to the HFT are curvedand rooted at the boundaries in such a way that one of theirfootpoints is close to the central region of a weak eld, whilethe other is far from it in the region of a strong eld (Fig.11d ). The turning pair of shearing motions makes the origi-nally distant footpoints even more distant (Fig. 11 e) byrotating and expanding the wholeHFT without a signicantcurrent concentration in it. On the contrary, the twistingpair of shearing motions brings the eld lines in the HFTcloser to each other by folding its original two-layer struc-ture into a one-layer structure (Fig. 11 f ). Thus, the resultingstructure becomes similar to the one obtained by Galsgaard& Nordlund (1996) in their braiding experiment (cf. Figs.11 f and 11 c). First, this parallel claries the cause of the cur-

Fig. 11. Schematic representation of the eld line variations as driving is imposed to the initial magnetic congurations ( a) and (d ). Top : Shows how theeld lines in Galsgaard & Nordlund(1996) experiment are braided by twosubsequent shearingboundary motions ( b)and( c) to produce thesituation ( c) wherethe current density grows exponentially with time. Bottom : Shows the situations for the experiments discussed in this paper. The varying thickness of the eldlines represents the corresponding variation of the magnetic eld strength. Being applied to the initially current-free HFT ( d ), the turning and twistingpairs of shearing motions produce the HFT deformations modeled in experiment turn and twist_1 [( e) and ( f ), respectively]. The fastest current growthin twist_1 is dueto a similarity of the eldline structures in ( c)and( f ).



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rent layer formation in the HFT by the twisting pair of shearing motions. Second, it explains why this process isdeveloping much slower than in the braiding experiment.Indeed, as it is seen from Figures 11 f and 11 c, the eld linesin the stressed HFT look much less bent than in its braidedanalogy, so that stressing of the HFT by the correspondingmagnetic tension forces also has to be much weaker. This, inturn, has to slow down the current accumulation in theHFT compared with the braided conguration, which is inaccord with our simulations.

Taking all this into account one can envision, with thehelp of Figure 12, how the pinching of HFT would occur fordifferent linear combinations of the perturbations depictedin panels (e) and ( f ). Shear corresponds to the perturba-tion of the type 12 e

12 f , which provides only a slight

accumulation of the current in the middle of the HFT. Inthe case of a 13 e

23 f , perturbation one would expect a

stronger HFT pinching to take place. Twist_4 conrmsthis assumption and shows, as expected, that the peak of thecurrent density appears in the plane z constant < 0:5closer to the boundary with the smaller shear. The perturba-tion of the type 23 e

13 f , however, will probably only

produce a current concentration on the boundary with thelarger shear but not in the volume.

Extending this approach a little further, we can anticipatethat a pure twisting motion rather than a twisting pair of shearing motions across HFT feet must be even more effec-tive for HFT pinching, because such boundary motionswould braid the eld lines in a way similar to the oneobtained by double-shearing motions (Fig. 11 c). In otherwords, the extra magnetic tension appearing in this case hasto stimulate a faster growth of the current density in themiddle of the HFT. To investigate this, two numericalexperiment with opposite vortex motions on the two drivingboundaries were carried out. Due to the periodicity of thenumerical domain the vortex motions have a limited size.

Therefore, in the rst experiment a form of the driving pro-le was chosen such that the sources had a nearly ridgedrotation introducing only a signicant shear in about 10%,measured in volume, of the sources farthest from the centerof rotation. For this case the magnetic ux responsible forthe current buildup along the HFT is insignicantly inu-enced by the rotational shear. As seen in the left panel of Figure 13, the twist of the HFT leads to a slow growth in thecurrent compared with the previously discussed shearexperiments. A second experiment with a high degree of rotational shear of the sources provides a much fastergrowth of the current with shear distance, where this dis-tance is measured relative the inner part of the ux tube

Fig. 13. Peak current in the plane halfway between the drivingboundaries as a function of rotational shear distance. The full line showsthe result from the ridged rotation, while the dashed line represents the casewitha large rotational shear of the sources.

Fig. 12. Velocity structure of turn and twist_1, with the background shading representing the magnitude of the current. The shading is scaled to themaximumminimumdynamic range in bothcasesthe current is 3.4 timesstronger in twist_1 thanin turn.



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(Fig. 13). Thus, the twisting component of boundarymotion across the HFT feet is responsible for its pinching,with the form of the rotational twist having asignicant inuence on the current growth.

In this connection, we can suggest a new interpretation of the results of Milano et al. (1999), who demonstrated theformation of a current layer between two coalescing twistedux tubes that are produced by the corresponding photo-spheric vortex motions. The two QSLs appearing betweenthese ux tubes are combined there in an HFT so that its feetare crossed by a twisting pair of shearing motions at thephotosphere. According to our consideration, this itself isalready enough for pinching of the HFT, while the energeti-cally favorable coalescence of the ux tubes may onlyenhance such a process by compressing the forming currentlayer in the correct direction.

5. CONCLUSIONS

The main objects studied here are hyperbolic ux tubes(HFTs) characterized by strong divergence of magnetic eldlines inside or, in other words, determined by large values of the squashing factor Q . We have made several numericalexperiments to investigate the importance of HFTs foraccumulating electric current. In these experiments a simpli-ed version of HFT was perturbed by imposing differentplasma ows at the photospheric boundary. It was shownthat the presence of an HFT in a given magnetic congura-tion is only one favorable condition for the current sheetformation process. As the second favorable condition, atwisting pattern of plasma ow at the HFT feet is required.This type of motion generates an electric current in the HFTpinching into a thin layer because of its interaction with theinitial hyperbolic structure of magnetic eld. Such a processis sustained by a stagnation ow in the middle part of theHFT, which is mostly subjected to the pinching deforma-tion. Thus, the twisting of an initially strongly squashed uxtube causes its strong pinching, which is actually a physical

essence of the current sheet formation process in topologi-cally simple three-dimensional magnetic congurations.

In this respect, the considered mechanism of current sheetformation is rather similar to the one investigated earlier byGalsgaard & Nordlund (1996) in their ux-braiding experi-ments. The necessary squashing of the initially uniform eldwas achieved there by shearing motion in one direction,while the necessary twisting of the eld was produced by sec-ond shearing motion in the perpendicular direction. Thecombination of these two motions resulted in a pinchingdeformation of the perturbed ux tube with a strong currentlayer in the middle.

In Paper I and in our present experiments, however,HFT appears as an inherent part of frequently observedquadrupole magnetic congurations, in which the required(for pinching) twisting ows across HFT feet result from thenatural motions of the sunspots that constitute the con-gurations. This circumstance enables us to consider thepinching of HFTs as the generic form of current sheetformation in aring quadrupole congurations.

In a more general context of magnetic energy release inthe solar corona we can also conclude that the prediction of the plausible sites of arelike activity requires one to knowboth HFTs themselves and the structure of the ow at theirfeet.

K. Galsgaard and T. Neukirch were supported by theParticle Physics and Astronomy Research Council(PPARC) in the form of Advanced Fellowships. Thenumerical MHD experiments were carried out usingPPARC funded Compaq MHD Cluster in Saint Andrews.This work was supported in part by the European Com-munitys Human Potential Programme under contractHPRN-CT-2000-00153, PLATON, and the contribution of V. S. Titov was supported by the Volkswagen Foundation.K. G. thanks the Niels Bohr Institute for Astronomy,Physics, and Geophysics for access to required facilitiesduring his visit.

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K. Galsgaard, V. S. Titov and T. Neukirch- Magnetic Pinching of Hyperbolic Flux Tubes. II. Dynamic Numerical Model