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Solar Physics (2004) 225: 267–292 C© Springer 2005

EFFECTS OF COMPLEXITY ON THE FLUX-TUBETECTONICS MODEL

R. M. CLOSE1, J. F. HEYVAERTS2 and E. R. PRIEST1

1School of Mathematics and Statistics, University of St. Andrews, St. Andrews, Fife,KY16 9SS, Scotland, U.K.

(e-mail: [email protected])2Universite Louis Pasteur, Observatoire de Strasbourg, 11 Rue de l’Universite,

67000 Strasbourg, France

(Received 25 July 2004; accepted 28 September 2004)

Abstract. The quiet-Sun magnetic field emerges through the solar photosphere in a multitude ofmixed-polarity magnetic concentrations and is subsequently tangled up into intricate regions of in-terconnecting flux. Moreover, since these discrete concentrations are likely to be extremely small insize, with fluxes of around only 1017 Mx, the number of such flux sources in, say, a supergranule, willbe extremely large. The flux-tube tectonics model of Priest, Heyvaerts, and Title (2002) demonstratedhow the formation and dissipation of current sheets along the separatrices that separate the regions ofdifferent connectivity are likely to make an important contribution to coronal heating. Since the fullcomplexity of the magnetic field is below present observable scales, this study examines the effect ofhaving the magnetic flux emerge through configurations structured on smaller and smaller scales. It isfound that, by fixing the amount of flux emerging into a given 2D region, the main factors influencingthe current build-up along the separatrices are the number of sources through which the flux emergesand the spatial distribution of the sources on the photosphere. The free energy (i.e., that above poten-tial) is stored lower and lower in the atmosphere as the complexity of the system increases. A simplecomparison is then made between coronal heating by separator currents and by separatrix currents. Itis found that both result in comparable amounts of energy release, with separatrix heating being themore dominant.

1. Introduction

Several different heating mechanisms are believed to be at work in the various mag-netic phenomena observed in the Sun’s outer atmosphere, the solar corona. In thequiet Sun, for instance, magnetic flux emerges through a multitude of mixed-polaritymagnetic concentrations (Livingston and Harvey, 1975; Zirin, 1987; Martin, 1988;Wang, 1988; Title, 2000; Hagenaar, 2001) that are dappled over the entire solarsurface (except in the active regions and sunspots). These concentrations, whichmove with granular and supergranular flows, are very dynamic in nature, with newconcentrations constantly emerging as older ones disappear. Furthermore, severalconcentrations may coalesce to form larger concentrations, while others divide intosmaller ones (Schrijver et al., 1997). The complicated motions of such concentra-tions in the photosphere will force coronal magnetic fields, with their footprintsrooted in these concentrations, to respond in different ways.

268 R. M. CLOSE, J. F. HEYVAERTS, AND E. R. PRIEST

Heating mechanisms tend to fall broadly into two main categories, dependingon the time-scale of the driving, i.e., heating by magnetic waves due to rapid driving(Hollweg, 1983; Goossens, 1991; Roberts, 1991) and heating by reconnection dueto slow driving (Parker, 1972; Heyvaerts and Priest, 1984). An example wherereconnection is important is the process of emergence of flux elements throughthe photospheric surface, where the emerging magnetic field reconnects with theoverlying field, as discussed by Close et al. (2004). In a similar manner, cancellationbetween collections of flux elements will also result in reconnection. This processhas been used to explain the presence of X-ray bright points in particular (Parnelland Priest, 1995).

Motions of the multitude of magnetic flux fragments in the solar photospherewill undoubtedly force the coronal magnetic field to react in a complex way. Sev-eral theories predict dissipation of magnetic energy at null points (Dungey, 1958;Schindler, Hesse, and Birn, 1988), whilst other models predict energy dissipationalong separators (Lau and Finn, 1990, 1996; Longcope, 1996, 1998; Priest and Titov,1996; Galsgaard and Nordlund, 1997; Galsgaard, Parnell, and Blaizot, 2000). Onemight also expect a certain amount of current to build up on the separatrix surfacesthat bound the complex domain structures that fill the Sun’s outer atmosphere. Re-cently, Galsgaard, Parnell, and Blaizot (2000), Priest, Heyvaerts, and Title (2002)and Mellor et al. (2004) also showed that dissipation may occur across separatrixsurfaces. Priest, Heyvaerts, and Title (2002) proposed a scenario, known as “flux-tube tectonics” by analogy with geophysical plate tectonics on Earth, wherebyreconnection is driven at the myriads of separatrix surfaces that separate regions ofinterconnectivity. They showed how simple lateral shearing motions (rather thancomplex braiding (Parker, 1979, 1994)) of separate flux sources lead directly to theformation of current sheets along the separatrices, which then dissipate rapidly byfast reconnection or in a turbulent manner.

Whilst high-resolution images from SOHO have revealed that small magneticfragments in the quiet Sun have typical fluxes of the order 1018 Mx, the likelihoodis that the fundamental units of flux in the quiet-Sun photosphere are a great dealsmaller, comprising flux tubes with a field strength of around 1200 G, diameter100 km and a flux of around 3 × 1017 Mx (Priest, Heyvaerts, and Title, 2002). Thiswould imply that each network element itself consists of around 10 of these intenseflux tubes, whilst ephemeral regions may contain as many as 100. This will haveimportant consequences for the field above the photosphere, due to the multitudeof unobserved flux tubes separated from one another by a great many separatrixsurfaces.

Here the aim is to model in 2 12 D the effect of photospheric flux dislocation on

coronal heating by adopting a standard model for 2 12 D magnetic fields used, for

example, by Zwingmann, Schindler, and Birn (1985), Wolfson and Low (1988),Amari and Aly (1990), Vekstein, Priest, and Amari (1990), Vekstein and Priest(1992, 1993) and Titov and Priest (1993). This is achieved by showing that the in-dependent motions of separate small 2D flux elements in the photosphere generate

EFFECTS OF COMPLEXITY ON THE FLUX-TUBE TECTONICS MODEL 269

current concentrations at the separatrix curves in the corona, where their flux be-comes more continuously distributed. The effects of complexity are examined byinvestigating what happens to the amount of free energy (i.e., that above potential)resulting from a small shear when the total absolute flux in a region is fixed and thenumber of concentrations through which it pierces the photospheric surface is var-ied. The next section, Section 2, introduces the problem and provides an overviewof two-dimensional fields. Section 3 then considers small perturbations to an initialpotential field configuration, with Section 4 providing a calculation of the extraenergy stored by this process. A simple four-source example is studied in Section5, whilst Section 6 discusses the effects of varying levels of complexity on theenergy stored. Section 7 provides a simple comparison between separator heatingand separatrix heating by revisiting the quadrupolar configuration of Section 5. Aconcluding discussion is then given in Section 8.

2. Flux Tube Tectonics

Priest, Heyvaerts, and Title (2002) considered a plane-stratified geometry with thephotosphere represented by the planes z = −L and z = L . By studying a config-uration in which the field emanates from two infinite arrays of uniformly spacedflux sources of equal magnitude, they found that in 2 1

2 D the shearing of fragmentedflux elements generates concentrated flux layers which, although not singular in theDirac sense, are nevertheless concentrated into extremely thin layers in the limitof large aspect ratios. In 3D, they found that the shearing motions give rise to adiscontinuity in the field across separatrices that supports a singular current sheetacross which the current density exhibits a Dirac distribution. In both cases, theyshowed that no significant current flows out with these sheets.

The work here is also carried out in Cartesian coordinates (x, y, z), assuming alocally flat photosphere in the plane y = 0. The magnetic field at the photosphere isassumed to be rooted in sources of negligible size that suffer relative motions withrespect to each other, driven by the photospheric flow. Initially, the coronal field istaken to be the potential field associated with these line sources.

Only simple motions are considered, in which the field sources are movedslightly by a shearing in the z-direction. The resulting coronal magnetic pertur-bation is calculated under the assumption that it can be treated by linearisationabout the potential field, and that the coronal magnetic structure remains at alltimes force-free.

2.1. OVERVIEW OF 2D MAGNETIC FIELDS

Following Low (1977), Zwingmann, Schindler, and Birn (1985) and many otherauthors, a 2 1

2 D field B(x, y) which is translationally invariant in the z-direction is


considered here. From the solenoidal property, it can be shown that such a fieldmay be expressed in terms of two functions a(x, y) and b(x, y) such that

B(x, y) = ∂a

∂yx − ∂a

∂xy + b z. (1)

It follows from B · ∇a = 0 that a(x, y) is constant along any field line. Muchuse will be made of this property throughout the subsequent analysis.

Ampere’s law gives the electric current density j, which, from Equation (1),takes the form

µ0j = ∂b

∂yx − ∂b

∂xy −

(∂2a

∂x2+ ∂2a

∂y2

)z. (2)

The Lorentz force can easily be calculated from Equations (1) and (2).

2.2. 2 12 D FORCE-FREE FIELDS

The coronal magnetic field is modeled here as force-free, so that

j × B = 0. (3)

For 2 12 D fields, the z-component of Equation (3) implies that b(x, y) is in fact a

function, b(a), of a(x, y), which in turn implies that b is constant on any given fieldline. The other components of Equation (3) reduce to a single equation for a(x, y)known as the Grad–Shafranoff equation which can be written, noting db/da =b′(a), as

�a + b(a)b′(a) = 0. (4)

3. Perturbation of 2D Potential Field into a 2 12 D Force-Free Field

Now suppose that an initially 2D potential field B0 with a vanishing z-componentis sheared by slightly moving, under perfect MHD conditions, the footpoints ofany field line in the z-direction. The relative footpoint displacement is of the formZ = Z z, with Z depending on the displacements of the sources to which the field lineis rooted. From the flux-freezing theorem, By(x, 0) is unchanged in such a motion,as is a(x, 0). The theorem also implies that plasma elements that were linked bya common field line in the initial configuration should remain so afterwards. Forexample, suppose a particular field line had its footpoints initially at (x1, 0, z0) and(x2, 0, z0), with z0 = 0 say. The field line’s footpoint at x1 moves from z0 = 0 toz1 = Z (x1), and the footpoint at x2 moves from z0 = 0 to z2 = Z (x2). The relativefootpoint displacement for this line is

Z12 = Z (x2) − Z (x1). (5)


In cases when Z12 is very small, the associated z-component of B will also beproportionally small, and the Grad–Shafranoff Equation (4) reduces, to first orderin Z12 (or b), to �a = 0, since bb′ is of second order. The boundary condition fora(x, 0) is unchanged and remains equal to a0(x, 0). Thus, the solution for a(x, y)after the infinitesimal shearing remains the same function, a0(x, y), as for the initialpotential field. The poloidal field B⊥ also remains unchanged and equal to B0.

The z-component of the field, b(a), is determined by the stretching experiencedby the corresponding field line. The general field-line equation reduces (in thepresent approximation) to

dx

B0x= dy

B0y= dz

b(a)= ds0

|B0| = ds0

|∇a| . (6)

Here ds0 denotes the line element along a field line of the poloidal field B⊥ = B0.By equating the third and fourth terms of Equation (6), an expression for Z12 maybe obtained by integrating along a poloidal field line (upon which a0(x, y) takesthe constant value a) from one foot point to the other:

Z12 = b(a)∫

a

ds0

|B0| . (7)

Relation (7) involves the specific volume V0 of the poloidal field line, definedby

V0 =∫

a

ds0

|B0| , (8)

so-named because the volume of a flux tube of small flux d� constructed about thisfield line is V0 d�. The electric current density j induced by such a small, perfectMHD stretching is given by

µ0j = b′(a)B⊥. (9)

3.1. PECULIARITIES OF FRAGMENTED-FLUX MODEL

In the model considered here, the distribution of flux on the photosphere (the planey = 0) enters the system via line sources parallel to the axis of invariance z. Thepotential field generated by a series of n such sources is given by the sum

B⊥(x, y) =n∑i

εi

π

r⊥ − ri

|r⊥ − ri|2 , (10)

where ri = xi x + yi y is the position of the i th source, with strength εi .Since in this model the flux remains constant between magnetic fragments, the

function a(x, 0) is constant between the fragments and suffers a discontinuous jumpby −εi when crossing a fragment i at ri, as illustrated in Figure 1. When

∑εi = 0,

one should expect that a(+∞, 0) = a(−∞, 0). Rotating about a source makes the


Figure 1. Example showing how a(x, 0) varies with x in a configuration consisting of four sources,with strengths εi (i = 1 . . . 4) as shown. Between pairs of sources a(x, 0) is constant and suffers adiscontinuous jump when passing over a source. For each source i , located at xi , the size of the jumpis equal to −εi .

Figure 2. Illustration of a point P lying at an angle θ on a semi circle of radius D (D � δ, the meansource separation), drawn about a source positioned at xi . At such a small distance from the source,the value of a(x, y) varies linearly about the circumference of the circle from a(xA, 0) at the point(xA, 0) to a(xB , 0) at the point (xB , 0).

identification of a easy, because very close to a source its own field is dominant,and B may therefore be approximated by only the field created by this source, sothat

B ≈ εi

π

r⊥ − ri

|r⊥ − ri|2 (11)

near source i , say. If a semi circle of radius D is taken about the source (shown inFigure 2), and the value of a(x, 0) at A is known, then at P , a(x, y) has the value

a(P) = a(A) + εi

πθ (P). (12)

This is only true if D is very small, so that field lines about the source i are stillstraight and radial at such a distance.

4. Extra Energy Stored

Free magnetic energy is stored in the coronal volume when the footpoints aresheared. Since the poloidal field remains unchanged for small motions, the change


in magnetic energy (per unit length in the z-direction) in such cases is essentiallythe energy of the displacement-aligned component

Wz =∫ ∫

b2

2µ0dxdy. (13)

Not all this energy is convertible into heat (Heyvaerts and Priest, 1984), butthis will nevertheless be assumed in order to judge the properties of energy storagedue to shearing. The surface element (dx, dy) is split as usual into dxdy ≡ d S =dsdσ⊥, where ds is a line element along the local potential field line and dσ⊥ isan infinitesimal element across it. The element dσ⊥ sustains a flux |da|, the twoquantities being related by |da| = |B⊥|dσ⊥. Since b depends only on a, this givesthe stored energy Wz as

Wz =∫ ∫

ds|da||B⊥

b2

2µ0=

∫da

b2(a)V0(a)

2µ0. (14)

In practice, this integral is approximated numerically by

Wz =∑i, j

Z2i j

2µ0

amax∫amin

da

V0(a). (15)

The integral can be calculated without difficulty because V −10 (a) is a regular

function which vanishes at the separatrix. Very interestingly, this shows that thosefield lines in a given cell linking two sources that contribute most to the energystorage are the lines of smallest specific volume. It will certainly emerge that muchenergy becomes stored in the ‘carpet’, i.e., regions where a lot of flux closes backdown in the form of short loops.

5. Example: Four Sources

In order to demonstrate the properties of the field following a translational shearingmotion, a case consisting of four sources, two positive and two negative (as shownin Figure 3(a)), is initially considered. If the sources were to represent networkfragments, then typical source fluxes, source separations and field strengths wouldbe 1018–1019 Mx, 14 Mm and 3 –30 G, respectively. In this simple scenario, thesources alternate in polarity and are spaced at uniform intervals. The magnitudesof all the strengths are equal, and a contour plot of the unperturbed field is shownin Figure 3(b). A plot of the specific volume about the second source from the leftis shown in Figure 4(a) as a function of a. The specific volume, in comparisonwith the values it takes close to the separatrix, is negligible in regions far from the


Figure 3. (a) Skeleton of the simple field arising from four sources positioned at −0.3, −0.15, 0.15and 0.3, with strengths 0.5, −0.5, 0.5 and −0.5, respectively, from left to right. The region studiedextends between −0.5 and +0.5 in the x-direction, and between 0 and 0.5 in the y-direction. (b)Contour plot of B2

0 for the initial potential field. Each contour level is a factor of 1.5 times theprevious level. The skeleton of the field is overplotted.

separatrix. In fact, V0 diverges at the separatrix, in the vicinity of which, as shownin Appendix A, it varies as

V0(a) ≈ L2

2|F | ln

(4|F |

|a − a∗|)

, (16)

where L is a characteristic length scale and F may be identified as F = 〈B∗0 〉L

(here 〈B∗0 〉 is a representative value of the field calculated about the null). As such,

F/L2 is therefore a gradient of the field about the null point.The two positive sources are then moved a distance of 5 × 10−4 units in the

negative z-direction, whilst the negative sources are moved a distance of 5 × 10−4

units in the positive z-direction. The resultant b component of the perturbed fieldabout the second source from the left is shown in Figure 4(b). Because V0(a)diverges on field lines passing close to the neutral point, the value of the perturbedcomponent of the field b(a) given by Equation (7) is very small on field lines passingclose to the neutral point, and is zero on the separatrices themselves.

The derivative of b(a) with respect to a is shown in Figure 4(c). Here it can beseen that the current given by Equation (9) is focused on the field lines that passvery close to the neutral point, and actually becomes infinite at the separatrix inthis model. Priest, Heyvaerts, and Title (2002) showed that this kind of behavior isgeneric for systems with translational symmetry.


Figure 4. (a) Specific volume V0(a) as a function of a, rotated about the negative source locatedfurthest to the left. Far from the region being examined, a(−∞, 0) = a(+∞, 0) = 0. (b) Resultingmagnitude of the field b in the z-direction, which is constant along each field line. The magnitude ofthe displacement field b is miniscule in comparison with the poloidal field |B0|– typically around 1%.(c) Derivative of b with respect to a; the current density j is given by j = (db/da)B0.


Figure 5. Contour plot of b2 for the sheared field. Each contour level is a factor of 4 times the previouslevel. The skeleton of the field is overplotted.

A contour plot of b2 is shown in Figure 5. Since the free energy stored (i.e., thatabove potential) scales as b2, it can be seen that the bulk of the energy is stored lowdown in the system.

6. Effects of Complexity

The properties of the current sheets that occur in a 2 12 D force-free field generated

by applying small shearing motions to a potential configuration have been demon-strated above. However, it should be expected that complexity in the distribution ofphotospheric fragments plays a role in determining macroscopic quantities, such asthe amount of energy stored and dissipated. Indeed, this complexity will affect, forexample, the total number of neutral points and separatrix surfaces in the configu-ration. Energy storage and dissipation may depend drastically on properties otherthan just the r.m.s. flux of the photospheric fragments or their average density in thephotosphere. They may depend, for example, on the statistics of the flux distributionamong fragments, their spatial clustering properties (or lack thereof), or the mixingof polarities on the photospheric boundary. Extrapolating further, they might alsodepend on the possibly fractal character of photospheric flux distribution (or lackthereof) and its effective fractal dimension.

A simple exploration of the effects that complexity in the photospheric fluxdistribution may have on the magnetic energy storage and coronal heating has beenundertaken here. Such a study, even for such a simplified model as that studied here,needs to be numerical. As such, limitations are placed on the distances betweensources and the range that their fluxes cover.

Energy storage has been computed for several models with varying spatial fluxdistributions on the x-axis. These models also differ in the number of fragmentsinvolved and the distribution of flux among the different fragments. For example, insome cases the fluxes of the fragments are all equal in magnitude, whereas in othercases the fluxes are drawn from chosen statistical distributions. Similarly, the spatialdistribution of fragments in space is in some cases taken to be deterministically


TABLE I

Summary of the source distributions described in the text. The ‘+’ and ‘−’signs represent the polarity of each source in sequence that is placed on thephotospheric axis. The ‘Clustered’ column indicates whether or not the sourcesare split into well-separated groups of four sources.

Source distribution Distribution pattern Clustered? Figures

(i) + + − − + + − − . . . No (a), (b)

(ii) + + − − + + − − . . . Yes (c), (d)

(iii) + − + − + − + − . . . No –

(iv) + − + − + − + − . . . Yes –

(v) Random No (e), (f)

(vi) Random Yes (g), (h)

simple and regular, whilst in other cases fragments are positioned randomly, and/orclustered to varying degrees. In all cases, the configuration is held in flux balance,with the total flux in the positive fragments equal in magnitude to the total fluxcontained in the negative fragments. In every run, the total absolute flux in thesystem is fixed at 2 arbitrary units.

The sources in each configuration have been placed between x = −0.3 andx = +0.3, with the region examined extending between x = −0.5 and x = +0.5.Table I summarizes concisely the parameters of the experiments that are performed.Six cases are considered, as follows:

(i) The sources are positioned regularly in space in sequences of two positivesources, followed by two negative sources, etc.

(ii) The sequence of the source signs is the same as case (i); however, this timesources are grouped together into clusters of 4.

(iii) The same as case (i), except that this time the sequence of the source signsvaries alternately from one source to the next.

(iv) The same as case (ii), but for the fact that the sequence of source signs alsovaries alternately from one source to the next.

(v) Half the sources are positive, and half are negative. They are scattered randomlythroughout the interval between x = −0.3 and x = 0.3.

(vi) Similar to case (v); however, the sources are clustered in groups of four con-taining two sources of each polarity. Each source is positioned randomly withinthe interval that its group occupies.

In the cases where fragments are positioned in a regular (unclustered) way,each consecutive pair of sources is separated by a distance 0.6/(n − 1), where n isthe number of sources in the given configuration. When sources are clustered intogroups of 4, the intervals in which the sources are placed are 1/3 of the separationbetween clusters.


In the following sections, several distributions of source strengths are consid-ered. These include a constant distribution, where the magnitudes of all the sourcesare equal, a uniform distribution, where all the source strengths are selected ran-domly with a uniform probability, an exponential distribution, and a power-lawdistribution.

6.1. SCENARIO 1: CONSTANT SOURCE STRENGTHS

In order to consider the effects of having the field emerge through a varying numberof sources, the number of sources in each case is initially set at eight. After havingcalculated the extra energy stored in such a field following a translational shearingmotion, the number of sources through which the flux emerges is increased to 12.In turn, this is increased to 16, and so forth, until the number of sources has reached200.

In the initial scenario, where all the sources are of equal magnitude, the cases inwhich the sources are spatially distributed in a symmetric way (cases 1–4) have, for aprescribed shearing motion, one unique value for the extra energy stored. However,in the cases where the sources are scattered randomly on the photospheric plane,there is an infinite number of possible fields. Thus, for a given number of sources,the extra energy stored is calculated for a sample of 50 fields.

Figure 6 shows plots of the energy stored versus the number of sources throughwhich the flux emerges. By examining Figures 6(a) and (c), and Figures 6(e)and (g), it is clear that the clustering of the sources is the most important fac-tor in affecting the amount of extra energy stored due to shearing motions; in thecases where the sources are clustered into groups of 4 (Figures 6(c) and (g)), thestored energy is significantly larger. The lines fitted to Figures 6(a), (c), (e) and (g)are given in Table II. By comparison, the differences in stored energy incurred dueto varying the sequence of the source signs on the photosphere, which invariablyreflects the density of neutral points on the photospheric plane, is fairly minimal.

The right-hand side of Figure 6 shows the fraction of the extra energy, W , storedbelow a given height, h (here h is normalized to total width of the region underexamination). It is clear that, in all cases, the height below which the majority ofthe energy is stored decreases as the number of sources increases. It is also clearfrom comparing Figures 6(b), (d), (f) and (h) that when there is a greater amountof energy, the height below which the majority of the flux is stored is lower.

In all the cases, the amount of stored energy increases linearly as the number ofsources is increased. This can be understood by carrying out a dimensional analysis.Considering a typical cell, the field is of the order

B0 ∼ �0

L0, (17)

where �0 is a typical source flux and L0 is typical length scale of an average cell.Of course, the field contains contributions from other sources at distances greater


Figure 6. Plots of the extra energy stored per unit length in the z-direction (left) for distributions inwhich all sources have equal strength, versus number of sources. The source positioning for each ofthe graphs is summarized in Table I. The fraction of the total energy stored in the system below agiven height is given on the right. In each plot, the rightmost curve represents the values obtained witheight sources; proceeding leftwards, subsequent curves give the values obtained with 12, 16, 20. . .

200 sources. Thus, as the number of sources increases, the energy becomes stored lower and lowerdown.


TABLE II

Equations fitted to the graphs on the left-hand side of Figure 6. yrepresents the extra energy stored and x represents the number ofsources through which the flux emerges.

Source distribution Figure Fitted line

(i) 6(a) y = 5.5x − 17.3

(ii) 6(c) y = 35x − 188.6

(iii) Similar to 6(a) y = 5.1x − 12.5

(iv) Similar to 6(c) y = 40.9x − 224

(v) 6(e) y = 14.5x + 8

(vi) 6(g) y = 137.9x − 815

than L0; however, close to the photosphere the field is largely dominated by thesources at the boundaries of the cell, so that, even if there are thousands of sourcesin the configuration, only a few have a non-negligible contribution. The specificvolume V0 may then be expressed as

V0 ∼ L0

B0∼ L2

0

�0. (18)

This means that the extra energy stored, Wz , scales as

Wz ∼ �0

V0∼ �2

0

L20

. (19)

If the number of sources is then increased by a factor of N from N to N N , thenthe flux �0 and length L0 change in the following way:

�0 → �0

N and L0 → L0

N . (20)

Clearly, as the number of sources is increased, the energy stored in a typical cellremains, on the whole, unchanged. However, the number of cells into which the fluxis partitioned does increase. As has already been seen, cells with long connections(which by their very nature are the ones that reach higher into the corona) do notcontribute much to the extra stored energy Wz . Thus, the majority of the extraenergy above potential comes from the cells that lie along the photospheric axis.In a two-dimensional configuration there is typically one such cell per source, sothat, when the number of sources is increased by a factor N , then the extra energyWz increases accordingly:

Wz → NWz. (21)

Hence Wz should be expected to increase linearly with N , which of course itdoes.


6.2. SCENARIO 2: RANDOM SOURCE STRENGTHS

Having studied the behavior due to sources that are all of constant magnitude, fieldsemanating from sources whose strengths are chosen at random are now analyzed.

In order to do this, three different source-strength distributions are considered; auniform distribution, an exponential distribution, and a power-law distribution. Inall these cases, the source strengths are selected randomly from the range [0.1, 1].In this way, it is ensured that there are no neutral points that lie too close to anyof the sources, which of course would make it difficult to resolve the problemcomputationally. The sources are then scaled so that the sum of the positive sourcesis 1, and the sum of the negative sources is −1.

For each of the 6 different spatial distributions, the experiment is performed overa sample of 50 fields, since randomly selecting source strengths means that there isan infinite number of fields, even when there is spatial symmetry in the distributionof the sources.

Figure 7 shows the results obtained using the uniform distributions (resultsobtained for the exponential and power-law distributions are similar). By comparingthe plots in all three figures, it can clearly be seen that the results obtained forthe various spatial distributions are all very similar. This naturally implies that thedistribution of source strengths has little effect on the extra energy stored due to theshearing motions. This is further backed up by looking at the equations of the linesfitting the plots, given in Table III. For a particular spatial distribution, the linesfitted to the data obtained from the three separate source-strength distributions arepractically the same.

7. A Comparison of Separator and Separatrix Heating

In this section, the four-source quadrupolar configuration of Section 5 is re-examined in order to make a simple comparison between heating resulting fromthe formation of current sheets upon separatrices and heating resulting from theformation of current sheets along separators.

7.1. 2D SEPARATOR CURRENT-SHEET FORMATION

Since the magnetic field in two dimensions is translationally symmetric, thesources are of infinite length in the direction of invariance (here taken to be thez-axis). This also means that the neutral point has an infinite extent in the z-direction,and is in fact a neutral line. In light of this, the calculation is restricted to a finiteportion of the system by placing conducting boundaries at the planes z = 0 andz = 1 (Longcope, 2001). Furthermore, in this 2D model, the role of the 3D separa-tor, which generally lies at the confluence of four separate magnetic domains, willbe played by the neutral line, which also lies at the confluence of four domains.


Figure 7. Plots of the extra energy stored per unit length in the z-direction (left) when source strengthsare selected from a uniform distribution, versus number of sources. The source positioning for eachof the graphs is summarized in Table I. The fraction of the total energy stored in the system belowa given height is given on the right. In each plot, the rightmost curve represents the values obtainedwith eight sources; proceeding leftwards, subsequent curves give the values obtained with 12, 16,20. . . 200 sources. Thus, as the number of sources increases, the energy becomes stored lower andlower down.


TABLE III

Equations fitted to data for the uniform source-strength distribution (Figure 7), the exponentialsource-strength distribution, and the power-law source-strength distribution. y represents the extraenergy stored and x represents the number of sources through which the flux emerges.

Source distribution Strength distribution Figure Fitted line

(i) Uniform 7(a) y = 5.5x − 16.3

(ii) Uniform 7(c) y = 34.3x − 184.4

(iii) Uniform Similar to 7(a) y = 5.7x − 14.1

(iv) Uniform Similar to 7(c) y = 44.5x − 246.8

(v) Uniform 7(e) y = 14.6x − 0.17

(vi) Uniform 7(g) y = 138.4x − 699.3

(i) Exponential Similar to 7(a) y = 5.6x − 20.4

(ii) Exponential Similar to 7(c) y = 34.2x − 195.3

(iii) Exponential Similar to 7(a) y = 5.9x − 16.8

(iv) Exponential Similar to 7(c) y = 44.7x − 254.5

(v) Exponential Similar to 7(e) y = 14.8x + 2.4

(vi) Exponential Similar to 7(g) y = 140.5x − 773.5

(i) Power Law Similar to 7(a) y = 5.6x − 13.9

(ii) Power Law Similar to 7(c) y = 34.2x − 170.2

(iii) Power Law Similar to 7(a) y = 6.2x − 15.7

(iv) Power Law Similar to 7(c) y = 44.4x − 236.7

(v) Power Law Similar to 7(e) y = 14.8x + 3.7

(vi) Power Law Similar to 7(g) y = 138.4x − 709.6

In this simple 2D scenario, illustrated in Figure 8(a), the four sources are linkedin all possible ways by the domains D1, D2, D3 and D4, which have fluxes ψ1,ψ2, ψ3 and ψ4, respectively. Since all source strengths are of equal magnitude, itturns out that ψ1 = ψ3 and ψ2 = ψ4. Displacing the sources along the x-axis ina vacuum magnetic field will in general lead to changes in the domain fluxes ψi

(i = 1 . . . 4). Following the displacement shown in Figure 8(a), domains D1 andD3 have their fluxes increased by an amount �ψ , whilst domains D2 and D4 havetheir fluxes decreased by the same amount. However, in the stick-slip reconnectionmodel of Longcope (1996), the field is assumed to thread a perfectly conductingplasma, which does not permit any flux to pass through the neutral point. Thus, acurrent ribbon with total current I forms along the neutral line with an orientationand self-flux such that it cancels �ψ . Longcope (2001) showed that the current Iand self-flux ψ (cr) are related by

ψ (cr)(I ) ≡ −�ψ(I ) = µ0 I

4πln

[16πe

I∗µ0|I |

], (22)


Figure 8. (a) Sketch of initial source configuration, with the directions in which the sources aredisplaced shown by arrows. The domain fluxes are shown by ψ1, ψ2, ψ3 and ψ4. (b) The configurationafter the source displacement. Domains D1 and D3 have had their fluxes increased by an amount �ψ ,whilst the fluxes of domains D2 and D4 have decreased by the same amount.

where I∗ is a characteristic current for the potential neutral point and is given byI∗ ≡ B ′Y 2

N (YN is the y-coordinate of the neutral point and B ′ is defined as |∇B0|calculated at the neutral point, with B0 denoting the initial vacuum field). Strictly,this expression is only valid in the vicinity of the neutral point (i.e., when |I | � I∗).The free energy above potential (�W ) stored by this current build-up is given by

�W = 12 I (�ψ)�ψ. (23)

Rather than allowing the current to build up indefinitely, the model of Longcope(1996) places a limit on the quantity I∗/|I |, so that, once such a threshold is reached,a ‘nanoflare’ is assumed to occur, in that an electric field parallel to the neutral lineensues and consequently allows the flux �ψ to pass through the neutral point. Itis also assumed that �ψ → 0, i.e., the field returns to its vacuum state, which isprecisely the potential field arising from the sources positioned as they are at theonset of the flaring event.

7.2. ANALOGOUS SEPARATRIX STICK-SLIP RECONNECTION

The current per unit length in the z-direction across a separatrix resulting from theshearing motions described in Section 3 follows from Equation (9), and is given by

I ′i =

∫Li

ji · dl =∫Ai

(dbi

da

)da, (24)

where Li follows the path of the separatrix i and Ai is across the separatrix. For thepurpose of making a comparison between the two heating models, a representative


current I ′∗i (i = 1 . . . 4) for each separatrix is defined as

I ′∗i = |Bmi |Li , (25)

where Li is the length of the separatrix curve i and Bmi is the value of the poloidalfield at the midpoint of the separatrix. Since the values of I∗ and I ′

∗i (i = 1 . . . 4) arenot directly comparable, a more universal criterion for comparing the two mecha-nisms must be found, hence the actual values of the various I ′

∗i are not too critical.In fact, due to the translational symmetry along the z-axis, all the I ′

∗i (i = 1 . . . 4)remain constant throughout the evolution of the system (unlike I∗, which changesits value after every event (i.e. release of energy)). It is assumed that when any oneof the four I ′

∗/|I ′i | reaches a threshold value, an instability occurs and the energy

stored in the system is released. Thus I ′i → 0 for all i .

7.3. COMPARISON OF ENERGY STORAGE

The evolution of the two configurations is managed such that each source is dis-placed by an amount �q. In the separator heating scenario, the sources are displacedalong the x-axis in accordance with the description in Subsection 7.1, while in theseparatrix heating scenario, the sources are sheared along the z-axis in accordancewith Section 5.

The value of �q is normalized to the initial distance between neighboring pairsof sources, and is varied between the limiting values of −0.5 and 0.5. In the case ofdisplacements along the x-axis, when �q = −0.5 the two central sources canceland the null point disappears; when �q = 0.5, the leftmost pair of sources cancels,as does the rightmost pair of sources, resulting in a magnetic void.

Figure 9. Extra energy stored (�E) as a function of source displacement (�q) for the two differentscenarios. The displacement axis is normalized to the initial separation between each of the sources.Extra energy stored following the shearing motions is given per unit length in the z-direction.


Figure 10. Extra energy stored (�E) as a function of source displacement (�q) for mean sourcedisplacements of (a) 0.02, (b) 0.04 and (c) 0.07 between events.

Figure 9 shows how the extra energy above potential would vary for the twotypes of source motions were the energy storage to increase unimpeded. The plotsuggests that energy storage associated with separatrix currents is more efficientthan energy storage associated with separator currents, with the ratio of separatrix


Figure 11. Plot showing how the total energy released by the system (�E) varies as a function ofthe mean source displacement between events.

to separator energy being typically 2– 4. It should be noted, however, that thevalues calculated for the larger displacements are less reliable, since the validityof Equations (15) and (23), which give the extra energy stored by the two systems,becomes somewhat over-stretched.

The three plots in Figure 10 show how the extra energy stored (�E) by the twoconfigurations varies with the mean displacement between flaring events. As onewould naturally expect, small source displacements between events (which are aconsequence of placing low thresholds on the values of the peak currents allowable)result in frequent releases of energy, and therefore not so much energy is storedbetween events as is stored when there are large displacements between the flaringevents. The question that then arises is: what releases a greater amount of energyin total – frequent releases of small bundles of energy, or less frequent releases ofrelatively large bundles of energy?

This question is addressed by Figure 11, which shows that in the both scenariosless frequent releases of energy result in a greater amount of energy being releasedin the long run. What is also shown is that, for a given mean source displacementbetween events, the total amount of energy released by the separatrix heating modelis around twice as much as that released by the separator heating model. Thissimple analysis suggests that separatrix heating and separator heating both makesignificant contributions to coronal heating, with separatrix heating being somewhatmore prevalent.

8. Conclusions

The effects of applying simple shearing motions to a series of line sources in atranslationally symmetric magnetic field have been studied here. It has been shownthat these motions lead to highly concentrated currents on the separatrices that arepresent in the system, with essentially no currents flowing far from them. The extra


energy generated by these shearing motions is stored very low down in the system,with little energy stored in the overlying upper coronal domains. In other words,the heating preferentially occurs near the footpoints of coronal loops, in agreementwith observations of TRACE loops.

By conserving the total amount of flux in the system, along with the lengthscales over which the flux emerges, it has been found that increasing the numberof sources through which the flux emerges results in an increased amount of storedenergy arising from the shearing motions. On the Sun, it is more natural to expectthe total absolute flux to increase as one considers smaller and smaller scales. This isbecause magnetogram images are comprised of a multitude of finitely sized pixels,meaning that opposite-polarity magnetic structures that exist on scales smaller thanthe dimensions of the pixel elements are likely to suffer from cancellation effects.Thus, all that is registered in a given magnetogram pixel is the net flux of the regioncovered by that pixel.

Nevertheless, this study has shown that the introduction of complexity alonemay greatly increase the energy storage of a given system. Such a discoveryis of particular relevance at the moment, where there is a growing realizationof the importance of fine-scale structures in the problem of how the corona isheated.

The results obtained here have natural implications regarding what should beexpected in three dimensions. Although the Grad-Shaffranov equation does not holdin 3D, making an analysis that is analogous to the work presented here extremelydifficult, one would expect to find results in 3D that are similar to those foundhere. (Indeed, Priest, Heyvaerts, and Title (2002) have already shown how currentsmay build up on 3D separatrix surfaces by considering a configuration comprisingpoint sources placed in a grid-style pattern on two infinite lattices at z = −L andz = L .) It would seem fairly intuitive to expect that the majority of the energy ina 3D system is also stored predominantly by the low-lying loops. However, it isshown in Appendix B that the amount of free energy arising from shearing motionsin 3D should be expected to fall as N as a linear decrease in the length scale isconsidered (although the amount of free energy arising from such shearing motionsfall as

√N when the number of sources increases linearly). This is because in the

example considered here, the number of sources is related linearly to the lengthscale; in 3D, there is an extra dimension to consider when placing sources on thephotosphere. This means that the number of sources should grow quadratically asthe length scale decreases, therefore affecting the amount of extra energy stored.

The simple comparison of separator heating and separatrix heating given here hassuggested that coronal heating associated with shearing motions and subsequentcurrent concentrations on separatrices play a substantial role in coronal heating,with indications that it may actually be more efficient at heating the corona thanheating associated with currents confined to separators. It will be interesting to seehow future work involving more complicated systems compares with the resultsfound here.


Acknowledgements

The authors would like to thank the Particle Physics and Astronomy Researchcouncil for financial support. We would also like to thank the European Commu-nity Human Potential Program Contract No. HPRN-CT-2000-00153, PLATON foradditional funding.

Appendix

A.1. SPECIfiC VOLUME ABOUT A 2D NULL POINT

Here an analytic expression for the value of V0 about a null point is obtained.In appropriate Cartesian coordinates, X and Y , say, a 2D potential field may berepresented locally in the vicinity of a null point by the flux function

a(X, Y ) = a∗ + F

L2(Y 2 − X2). (A.1)

The magnetic field which corresponds to this is

Bx = 2F

L2Y, By = 2

F

L2X, (A.2)

where the X and Y axes are oriented such that locally about the origin the field isthat of a classical X-type neutral point. The quantity F/L2 may then be physicallyidentified as being related to the gradient of the field near the null point:

2F

L2= d Bx

dY= d By

d X= ±

∣∣∣∣∇√

B2x + B2

y

∣∣∣∣. (A.3)

This shows that near the null point |B| depends only on the distance from thenull point. Hence F/L2 describes the structure of the magnetic field near the nullpoint. Letting δa be the difference in the flux function between a given non-singularfield line a and the flux function of the separatrix a∗ gives

δa = a − a∗. (A.4)

In the vicinity of the neutral point, this may be rewritten as

δa = F

L2(Y 2 − X2). (A.5)

Now consider a section of a field line passing the Y axis at Y0 and extendingfrom −X to +X . Next, consider how the specific volume of this section varies asthe separatrix is approached, i.e., when |δa| approaches zero. The value of δa forthis line is

δa = F

L2Y 2

0 , (A.6)


and a line element along this field line is

ds2 = d X2 + dY 2 = d X2

(X2

Y 2+ 1

). (A.7)

With the modulus of the field being given as

|B| =(

4

(F

L2

)2

(X2 + Y 2)

) 12

, (A.8)

this leads to

ds

|B| = L2d X

2|F |

√X2

Y 2 + 1√

X2 + Y 2= L2d X

2|FY | = L2d X

2|F |√

Y 20 + X2

. (A.9)

The specific volume of the line between ±X is then

V 0 =∫ +X

−X

L2d X

2|FY0|√

1 + X2/Y02

= L2

|F | arg sinh

(X

Y0

), (A.10)

which can be expressed in terms of δa, the difference in the flux function acrossthe separatrix, as

V 0 = L2

|F | arg sinh

(X

√|F |√L2

√δa

). (A.11)

For very small δa, the approximation sinh(u) ≈ ln(2u), which is valid for largeu, may be used. This leads to the expression

V 0 = L2

|F | ln

(2X

√|F |√L2

√δa

)≈ L2

2|F | ln

(4|F ||δa|

). (A.12)

A.2. ENERGY STORAGE IN A 3D CONFIGURATION

It was shown in Subsection 6.1 that the energy stored in a 2 12 D configuration over a

fixed region following a simple shearing motion grows linearly with the number ofsources through which a given amount of flux emerges. Here, similar dimensionalarguments are used to explore what happens in 3D. Assuming that the initiallypotential magnetic field arises from a series of sources which lie in a 2D plane andare subjected to a shearing motion, it is further assumed that the overlying coronaconsequently suffers a quasi-static, perfect MHD change. This turns the potentialmagnetic configuration into one that is force-free, associated with a displacementfield ξ . The resulting field perturbation is

b = ∇ × (ξ × B0). (A.13)


The field in a typical 3D magnetic domain is of the order

B0 ∼ �0

L20

, (A.14)

where �0 is a typical source flux and L0 is a typical length scale of an averagedomain. As in 2 1

2 D, the field contains contributions from other sources at distancesgreater than L0; however, close to the photosphere the field is largely dominatedby the sources on the boundaries of the domain, so that, even if there are thousandsof sources in the configuration, only a few have a non-negligible contribution. Atypical field perturbation may then be seen to be of the order

b0 ∼ �0

L20

. (A.15)

The energy associated with the field perturbation in a typical domain may beexpressed as

Wb ∼ b20 L3

0 ∼ �20

L0. (A.16)

If the number of sources N is then increased by a factor of N to N N , then theflux and length scales change in the following way:

�0 → �0

N and L0 → L0

N 12

. (A.17)

This implies that the energy stored in a typical domain varies as Wb → Wb/N32 .

As in 2D, though, the number of domains increases as the number of sourcesis increased. Furthermore, the majority of the extra energy stored in the systemfollowing the source displacement is likely to be stored in the low-lying domains.There are typically 3 – 4 planar domains per source in 3D (Schrijver and Title, 2002)(planar domains are domains that have field lines lying in the source plane). Thus,the total energy stored in the system, Wb, as a result of the source displacementvaries as

Wb → Wb

N 12

. (A.18)

It may also be shown in a similar way that the total energy stored in the systemfollowing a fixed shearing motion decreases linearly with a linear decrease in thelength scale L0 (i.e., Wb → Wb/N for L0 → L0/N ).

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