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GENERALIZED SQUASHING FACTORS

FOR COVARIANT DESCRIPTION OF

MAGNETIC CONNECTIVITY IN THE SOLAR CORONA

V. S. Titov

SAIC, 10260 Campus Point Drive, San Diego, CA 92121-1578

titovv@saic.com

ABSTRACT

The study of magnetic connectivity in the solar corona reveals a need to generalize the field linemapping technique to arbitrary geometry of the boundaries and systems of coordinates. Indeed,the global description of the connectivity in the corona requires the use of the photospheric andsolar wind boundaries. Both are closed surfaces and therefore do not admit a global regularsystem of coordinates. At least two overlapping regular systems of coordinates for each of theboundaries are necessary in this case to avoid spherical-pole-like singularities in the coordinates ofthe footpoints. This implies that the basic characteristic of magnetic connectivity—the squashingdegree or factor Q of elemental flux tubes (Titov et al. 2002)—must be rewritten in covariantform. Such a covariant expression of Q is derived in this work. The derived expression is veryflexible and highly efficient for describing the global magnetic connectivity in the solar corona. Inaddition, a general expression for a new characteristic Q⊥ which defines a squashing of the fluxtubes in the directions perpendicular to the field lines is determined. This new quantity makes itpossible to filter out the quasi-separatrix layers whose large values of Q are caused by a projectioneffect at the field lines nearly touching the photosphere. Thus, the value Q⊥ provides a much moreprecise description of the volumetric properties of the magnetic field structure. The differencebetween Q and Q⊥ is illustrated by comparing their distributions for two configurations, one ofwhich is the Titov-Demoulin (1999) model of a twisted magnetic field.

Subject headings: Sun: coronal mass ejections (SMEs)—Sun: flares—Sun: magnetic fields

1. INTRODUCTION

The structure of magnetic field is often animportant factor in many energetic processes inthe solar corona. This especially refers to thetopological features of magnetic structure suchas null points, separatrix surfaces, and separa-tor field lines. They serve as preferred sites forthe formation of current sheets and the corre-sponding accumulation of the free magnetic energy(Sweet 1969; Baum & Bratenahl 1980; Syrovatskii1981; Lau & Finn 1990; Longcope & Cowley 1996;Priest & Titov 1996; Priest & Forbes 2000; Longcope2001). The magnetic reconnection process inducedin the current sheets at some critical parametersallows the accumulated magnetic energy to con-

vert into other forms: thermal, radiative and ki-netic energy of plasma and accelerated particles.This process is considered to be a driving mech-anism of many energetic phenomena in the solaratmosphere (Priest & Forbes 2000; Parker 1979,1994).

Over the last decade, it also became clearthat the geometrical analogs of the separatrices(Longcope & Strauss 1994a,b; Titov et al. 1999;Titov & Hornig 2002; Titov et al. 2002), the so-called quasi-separatrix layers (QSLs, (Priest & Demoulin1995; Demoulin et al. 1996a, 1997)), have similarproperties. There are indications that the QSLsare probably more ubiquitous than the true sep-aratrices (Titov et al. 2002). This increases the

1

significance of the problem of determining QSLsin a given magnetic configuration. In comparisonwith the separatrices, the determining of QSLsrequires a more sophisticated technique, whichis based on a point-wise analysis of the mag-netic field line connectivity. The basic quantityin this technique is the squashing degree or fac-tor Q of elemental magnetic flux tubes. Thisquantity has previously been defined for the pla-nar geometry (Titov et al. 1999; Titov & Hornig2002; Titov et al. 2002), which provides a good ap-proximation for describing magnetic structures inactive regions with the characteristic size smallerthan the solar radius R⊙.

Such an approximation, however, is hardly ap-plicable for a global description of magnetic con-nectivity in the solar corona including the openmagnetic field of the coronal holes. The corre-sponding large-scale structure of magnetic fieldsis also of substantial interest for solar physics, es-pecially, for understanding solar eruptions. So therespective generalization of the above techniquemust allow us to determine Q for the coronal vol-ume bounded by the photospheric and solar-windsurfaces. This immediately raises technical prob-lems, which do not exist in the case of the pla-nar geometry. First, both these boundary surfacesare closed, and therefore, none of them admits aglobal regular system of coordinates. To avoid acoordinate singularity of a spherical-pole type, atleast two overlapping coordinate systems (coordi-nate charts) must be used in this case for describ-ing the locations of the field line footpoints on eachof the boundries. Second, the solar-wind bound-ary surface generally cannot be a sphere, but someother curvilinear surface whose geometry dependson the coronal magnetic field (Levine et al. 1982).These two requirements of the technique can besatisfied only by using a covariant approach tothe description of Q with the coordinate systemsthat are generally different for each of the bound-aries. The derivation of such a covariant expres-sion for the squashing factor is one of the goals ofthe present work.

The second goal of the work is to make an es-sential refinement of the squashing factor itself.The problem is that the large values of Q maybe caused not only by squashing of elemental fluxtubes in the volume but also by a projection effectat the boundary surfaces. The latter occurs at the

field lines which are nearly touching the boundaryat least at one of the footpoints. This effect, inparticular, takes place in the vicinity of the baldpatches (BPs, (Titov et al. 1993)), which are thesegments of the photospheric polarity inversionline (PIL), where the coronal field lines touch thephotosphere. When analyzing magnetic connec-tivity, it is important to discriminate between theprojection effect and volumetric squashing. Forthis purpose, we derive a covariant expression forthe perpendicular squashing factor Q⊥, which de-scribes the squashing of elemental flux tubes onlyin the directions orthogonal to the field lines.

Sections 2 and 3 present the derivations of Qand Q⊥ and demonstrate on the examples of pla-nar and spherical geometry how to apply thesegeneral expressions. The difference between Qand Q⊥ is considered in detail in §4 by calculatingand comparing these quantities for two particularmagnetic configurations. The obtained results aresummarized in §5.

2. COVARIANT FORM OF THE

SQUASHING FACTOR

Consider a plasma-magnetic configuration ina finite volume with a smooth boundary of anarbitrary shape. It may generally consist oftwo or even more surfaces,—for example, thephotosphere and the solar-wind surface form aboundary for the entire solar corona. Each ofthe two footpoints of a given magnetic field linemay belong in general to any of these surfaces.We will use the designations “launch” and “tar-get” for the footpoints and parts of the bound-ary surfaces at which the field lines start andend up. Let (u1, u2) and (w1, w2) be the sys-tems of curvilinear coordinates at the launch andtarget boundaries, respectively. The magneticfield lines connecting these boundaries define amapping (u1, u2) → (w1, w2) determined by somevector-function (W 1(u1, u2),W 2(u1, u2)). The lo-cal properties of this mapping are described bythe Jacobian matrix

D =

[

∂W i

∂uj

]

. (1)

For each field line, this matrix determines a lin-ear mapping from the tangent plane at the launchfootpoint to the tangent plane at the target foot-point, so that a circle in the first plane is mapped

2

into an ellipse in the second plane (Fig. 1a).The aspect ratio of such an ellipse defines the de-gree of a local squashing of elemental flux tubes,which means that any infinitesimal circle centeredat a given launch point is mapped along the fieldlines into an infinitesimal ellipse with this as-pect ratio at the target footpoint. This gener-alizes a coordinate-free definition of the squash-ing factor to the case of curvilinear boundaries,whose tangent planes are generally not the same,as is in the case of plane boundaries consideredin Titov et al. (1999); Titov & Hornig (2002) andTitov et al. (2002).

To derive an analytical expression for the as-pect ratio of the above ellipse, let us introducefirst a vector function R(u1, u2) that describes ina three-dimensional (3D) Cartesian system of co-ordinates the locations of the footpoints at thelaunch boundary. Then the vectors

ǫk =∂R

∂uk, k = 1, 2, (2)

determine at this boundary the covariant vectorbasis tangent to the u-coordinate lines. Thus,

glk = ǫl·ǫk, l, k = 1, 2, (3)

is the corresponding covariant metric tensor,which determines local lengths and angles at thelaunch boundary. The dot here stands for theusual scalar product in 3D Euclidean space.

Using (2) and the standard Gramm-Schmidtprocedure, one can construct an orthonormal basis

e1 =ǫ1√g11

, (4)

e2 =g12√g g11

ǫ1 −√

g11g

ǫ2. (5)

Hereafter g ≡ det [glk] is a determinant of the co-

variant metric tensor. Now, any point of a circleof unit radius in the plane tangent to the launchboundary is represented by the vector

o = cosϑ e1 + sinϑ e2, (6)

whose angle parameter ϑ ∈ [0, 2π).

Suppose that the vector-function R(w1, w2) de-fines the points at the target boundary, then

o = ok∂W i

∂ukǫi (7)

Fig. 1.— Linearized field line mapping of a cir-cle into an ellipse between tangent planes at thelaunch and target boundaries (a) and between theplanes perpendicular to the field line at its foot-points (b). In general, different and arbitrary coor-dinates (u1, u2) and (w1, w2) with covariant bases(ǫ1, ǫ2) and (ǫ1, ǫ2), respectively, are assumed atthe launch and target boundaries.

3

is the field-line mapping image of o at the tan-gent plane of the target boundary, where the cor-responding covariant basis vectors, parallel to thew-coordinate lines, are

ǫi =∂R

∂wi. (8)

Hereafter a summation over repeating indices withtheir values running from 1 to 2 is assumed.

With varying ϑ, the vector o traces in this planean ellipse such that

o2 ≡ g∗ij o

ioj =1

2g∗ij

∂W i

∂uk

∂W j

∂ul

×[

ek1el1 + ek2e

l2 + cos 2ϑ

(

ek1el1 − ek2e

l2

)

+sin 2ϑ(

ek1el2 + el1e

k2

)]

, (9)

where the asterisk indicates that g∗ij(u1, u2) is a

result of evaluating gij(w1, w2) at the target foot-

point (w1, w2) = (W 1(u1, u2),W 2(u1, u2)).

After some simple trigonometry and lengthy al-gebra using equations (2)–(5), equation (9) is re-duced to

o2 =

1

2(N2 +

√

N4 − 4∆2 sin 2ϑ), (10)

where N2 and ∆ are determined by

N2 =∂W i

∂ukg∗ij

∂W j

∂ulglk, (11)

∆ =

√

g∗

g

∂(W 1,W 2)

∂(u1, u2), (12)

in which g and g∗ denote the determinants of co-variant metric tensors at the launch and targetfootpoints, respectively. The components of thecontravariant metric tensor glk can be viewed hereas elements of the inverted matrix [glk]

−1of the

covariant metric. The value ϑ is simply ϑ plus anadditional value, which is independent of ϑ andwhose expression does not matter for the presentconsideration.

What actually matters is that sin 2ϑ runs valuesfrom −1 to +1 when o(ϑ) and o(ϑ) are tracing,respectively, the above circle and ellipse. The min-imum−1 and maximum +1 correspond here to theminor and major axes of the ellipse, respectively,so that its aspect ratio is

omax

omin

=

(

N2 +√N4 − 4∆2

N2 −√N4 − 4∆2

)1/2

=N2

2|∆| +

√

(

N2

2|∆|

)2

− 1. (13)

The large values of this ratio do not differ muchfrom its asymptotic value

Q = N2/|∆|. (14)

Note also that Q ≥ 2, since inverting equa-tion (13) yields Q = omax/omin + omin/omax andomax/omin ≥ 1. Therefore equation (14) will beused as a covariant definition of the squashing fac-tor.

It is evident from the derivation of Q that thisvalue is invariant to the direction of field line map-ping. Indeed, the inverse mapping implies locallythat 1/omin is a maximum stretching coefficientand 1/omax is a minimum shrinking coefficient.Such coefficients will coincide with the lengths ofthe major and minor axes of the ellipse obtainedfrom a circle of a unit radius due to this inversemapping. Thus, although this new ellipse has dif-ferent lengths of axes, their ratio is the same asfor the previous one, which proves the statement.A formal proof of the statement is also not diffi-cult to obtain by using the derived expressions ofN2 and ∆ in a similar way as in the case of planeboundaries (Titov et al. 1999). The invariancy ofQ to the direction of field line mapping justifiesits status of a correct measure for the magneticconnectivity.

Note also that ∆ for a given infinitesimal fluxtube is a ratio of its cross section areas at the tar-get and launch points. Therefore, since the mag-netic flux is conserved along the tubes, this valuecoincides with the corresponding inverse ratio ofthe normal field components, so that

∆ = Bn/B∗n, (15)

where Bn and B∗n are normal components of the

magnetic field to the boundaries at the conjugatelaunch and target footpoints. In practice, the nu-merical calculation of ∆ through this ratio is moreprecise than that given by equation (12) and there-fore it should be used for computing ∆ in equation(14).

The above mathematical construction is re-lated to the Cauchy-Green deformation tensor(Marsden et al. 2002) known in the theory of elas-

4

ticity. It can be written in our notations as

Ckl =∂W i

∂ukg∗ij

∂W j

∂ul, (16)

where (W 1(u1, u2),W 2(u1, u2)) and gij represent,respectively, a finite deformation and covariantmetric tensor of an elastic two-dimensional body.The contraction of the Cauchy-Green tensor witha pair of orthonormal vectors em and en yields thetensor

Cmn = Cklekmeln (17)

such that its eigenvalues coincide with the squaredsemiaxes o2max and o2min of the above ellipse. Thesquare root of their ratio defines in accordancewith (13) and (14) the squashing factor Q.

It should be emphasized that this analogy ispossible only in our general approach, where twoindependent systems of coordinates are used fordescribing the location of the conjugate footpoints.This allows us to apply coordinate transforma-tions only at the launch boundary, while keepingthe coordinates at the target boundary unchanged.With respect to these transformations, the objectdefined by (16) does behave as a covariant second-rank tensor. The latter is not valid, however, ifone global 3D system of coordinates is used for de-scribing the entire field configuration and so bothboundaries are subject then to coordinate trans-formations.

This has only a methodological meaning anddoes not exclude, of course, an application of thederived expressions to such particular cases. Forexample, consider a closed magnetic configura-tion in the half space x3 ≥ 0 with the globalCartesian coordinates (x1, x2, x3) ≡ (u1, u2, x3) ≡(w1, w2, x3) and the photospheric boundary planex3 = 0. The field line mapping is then given by(

X1(x1, x2), X2(x1, x2))

. There are no more dif-ferences between upper and low indices and con-travariant glk and covariant g∗ij metrics; the latter

simply turn into Kronecker symbols δkl and δij .So equations (11), (12) and (15) are reduced to

N2 =∂X i

∂xk

∂X i

∂xk, (18)

∆ =∂(X1, X2)

∂(x1, x2)=

B3

B∗3

, (19)

as required in this case (Titov et al. 2002).

Consider now a more complicated class of con-figurations, where both open and closed mag-netic field lines are present. Let the configurationbe described in one global system of coordinates(r, θ, φ), where r = R⊙ corresponds to the pho-tospheric launch boundary, while r = R∗ repre-sents the target boundary. For the open field linesreaching the spherical solar-wind boundary of ra-dius RSW, we put R∗ = RSW, while for the closedones we take R∗ = R⊙. Thus, u1 = w1 = φ,u2 = w2 = θ and the field line mapping is(Φ(φ, θ),Θ(φ, θ)), which yields

[g∗ij ] =

(

R2∗ sin

2 Θ 00 R2

∗

)

(20)

and

[glk] =

(

R−2⊙ sin−2 θ 0

0 R−2⊙

)

, (21)

where the contravariant metric glk at the launchboundary is obtained from the corresponding co-variant metric by inverting it simply as a 2×2 ma-trix. Using these expressions and equation (11),we obtain

N2 =R2

∗

R2⊙

[

(

sinΘ

sin θ

∂Φ

∂φ

)2

+

(

sinΘ∂Φ

∂θ

)2

+

(

1

sin θ

∂Θ

∂φ

)2

+

(

∂Θ

∂θ

)2]

. (22)

Equations (12) and (15) in this case become

∆ =R2

∗

R2⊙

sinΘ

sin θ

∂(Φ,Θ)

∂(φ, θ)=

Br

B∗r

. (23)

The obtained expressions for N2 and ∆ haveseeming singularities at the poles, where they ac-tually reduce in the generic case to resolved inde-terminacies with Φ(φ, θ) and Θ(φ, θ) proportionalto sin θ. These indeterminacies are artificial andunrelated to some special properties of the mag-netic structure; they are caused by the pole singu-larities inherent in the used global spherical sys-tem of coordinates. Moreover, their appearanceis unavoidable in any other global system of co-ordinates on a closed sphere-like surface becauseof its intrinsic topological properties. Thereforethis may generally reduce precision of a numericalevaluation of Q near the pole-like points. To avoid

5

such indeterminacies, at least two overlapping co-ordinate charts on each of the spherical boundariesare required. For this purpose, it is sufficient touse two systems of spherical coordinates turnedwith respect to each other on the right angle inthe θ-direction. Switching from one of such sys-tems in its polar regions to the other, as suggested,for example, by Kageyama & Sato (2004), makesit possible to resolve the problem of the pole in-determinacies. The required expressions for suchcalculations of Q can be obtained again from equa-tions (11) and (12) with properly modified metrictensors.

3. PERPENDICULAR COVARIANT

SQUASHING FACTOR

To find the perpendicular squashing factor Q⊥,we need to know the field line mapping betweeninfinitesimal planes orthogonal to the field linesat the conjugate footpoints (Fig. 1b). Note firstthat the projection effect at the boundaries is lo-cal, because it depends only on the orientations ofthe tangent planes at the boundaries with respectto the vectors of magnetic field at the footpoints.So the required mapping between the indicatedorthogonal planes can be obtained from the re-spective mapping between the tangent planes bycorrecting it only at such footpoints. This impliesthat Q⊥ can be expressed in terms of the samevalues as Q and, in addition, the field vectors B

and B, respectively, at the launch and target foot-points.

We will derive Q⊥ by using the same procedureas for Q while modifying it in accordance with theabove comments. The vector o tracing the circleof unit radius is given by the same expression (6).However, since it lies now in the plane perpen-dicular to the field line at the launch point, thecorresponding orthonormal basis is chosen to beorthogonal to B, so that

e1 =B×ǫ1

|B×ǫ1|, (24)

e2 =B×e1

|B×e1|. (25)

This vector o is mapped along the field lines into avector o lying in the plane perpendicular to the lo-cal field B at the target footpoint. The respectivemapping dW⊥ can be represented as a composi-

tion P−1 ◦dW ◦P of three others according to thefollowing diagram:

odW⊥

−→ o

↓P ↑P−1

odW

−→ o

(26)

Here the mapping P projects the vector o alongBonto the plane tangential to the launch boundaryto yield the vector

o = o− (o · ǫ1×ǫ2)

(B · ǫ1×ǫ2)B, (27)

which has a vanishing component along the vec-tor ǫ1×ǫ2 perpendicular to such a plane. Thenthis vector o is mapped by the differential of thefield line mapping dW determined by the Jacobianmatrix (1) into the vector

o = ok∂W i

∂ukǫi (28)

which lies in the plane tangential to the targetboundary. Finally, the obtained vector o is pro-jected by P−1 along B at the target footpoint ontothe plane perpendicular to B to result in

o = o−(

B·o)

B2B. (29)

Eliminating now o and o from (27)–(29) andusing (6) with the basis from equations (24)–(25),we express o in terms of ǫ-basis and magnetic fieldvectors at the conjugate footpoints. This allows usto calculate o

2 and then Q⊥ in a similar way asdone before when deriving Q. The result is

Q⊥ = N2⊥/|∆⊥|, (30)

N2⊥ =

∂W i

∂ukg∗⊥ij

∂W j

∂ulg⊥lk, (31)

∆⊥ =

√

g∗⊥g⊥

∂(W 1,W 2)

∂(u1, u2), (32)

where the asterisk has the same meaning as inequations (9), (11) and (12). Thus, the obtainedQ⊥ differs from Q only in the form of the metrictensors, which are determined now by

g∗⊥ij =

(

gij −BiBj

B2

)∗

, (33)

g⊥lk = glk +gBlBk

(B · ǫ1 × ǫ2)2. (34)

6

The asterisk here implies automatically that thecorresponding values refer to the target footpoint,therefore the tilde used for indicating this fact inintermediate equation (29) is omitted in the finalexpression (33). One can also check that

[

g⊥lk]

=

[g⊥lk]−1

and

g⊥ ≡ det [g⊥lk] =(B · ǫ1 × ǫ2)

2

B2, (35)

so that equation (32) reduces to

|∆⊥| =|B||B∗| (36)

if, in addition, the magnetic flux conservation istaken into account. This expression should beused instead of (32) for computing |∆⊥| in (30)for the same reason that (15) should be used in(14). Thus, formulas (30)–(36) completely definethe covariant expression for Q⊥.

Let us see now how these formulas work inthe case of a closed magnetic configuration de-scribed in a global Cartesian system of coordinates(x1, x2, x3) ≡ (u1, u2, x3) ≡ (w1, w2, x3) with thephotospheric boundary plane x3 = 0. The formu-las are significantly simplified in this case to yield

N2⊥ =

∂X i

∂xk

(

δij −B∗

i B∗j

B∗2

)

×∂Xj

∂xl

(

δlk +BlBk

(B3)2

)

, (37)

where the values of the corresponding covariantand contravariant components of vectors and ten-sors do not differ from each other.

Note also that expression (37) apparently di-verges near the PIL, where the normal field com-ponent B3 vanishes. This is actually not a truesingularity but rather an indeterminacy, which isresolved to give a low limit of Q⊥

∣

∣

PIL= 2 if the

PIL has no BPs. At the BPs, such an indeter-minacy is also resolved but it may generally havedifferent limits ofQ⊥ > 2 at the left and right sidesof BPs. Thus, Q⊥ may experience a jump whencrossing BPs, unless the configuration is symmet-ric as in the example of §4.2.

Similar to §2, consider also a more general classof configurations, where both open and closed fieldlines are present and bounded by a spherical solar-wind surface of radius RSW and the photosphere

of radius R⊙ as before. It is convenient in this caseto use matrix notations, in which equation (31) iswritten as

N2⊥ = tr

(

DTG∗⊥DG⊥

)

, (38)

where

D =

∂Φ

∂φ

∂Φ

∂θ∂Θ

∂φ

∂Θ

∂θ

(39)

is the Jacobian matrix of the field line mapping.The covariant and contravariant metrics at thetarget and launch boundaries, respectively, are de-termined by the following matrices:

G∗⊥ ≡ [g⊥ij ]

∗

= R2∗

sin2 Θ

(

1−B∗2

φ

B∗2

)

− sinΘB∗

φB∗θ

B∗2

− sinΘB∗

φB∗θ

B∗21− B∗2

θ

B∗2

, (40)

G⊥ ≡[

g⊥lk]

= R−2⊙

sin−2 θ

(

1 +B2

φ

B2r

)

BφBθ

sin θ B2r

BφBθ

sin θ B2r

1 +B2

θ

B2r

, (41)

where R∗ = R⊙ if a given footpoint belongs to aclosed field line and R∗ = RSW otherwise.

The matrix G⊥ has two types of singularities,which actually lead only to indeterminacies in ex-pression (38). The first indeterminacies take placeat the poles of spherical coordinates (θ = 0, π).They are already discussed above in connectionwith equation (22). The second indeterminaciestake place at the PIL, where Br = 0 . They areresolved in much the same way as occurred in theprevious case of Cartesian geometry to give a fi-nite value of Q⊥

∣

∣

PIL. If the PIL has no BPs, the

length of the field lines vanishes near the PIL, sothat D → I and G∗

⊥ → G⊥−1, which results inN2

⊥ → 2. The presence of BPs implies, however,a strong projection effect in their neighborhoodand the corresponding singularities in the Jaco-bian matrix D at the BPs. These singularities areexactly of the same type and value as those in G⊥

but opposite in signs. So they cancel each other

7

in equation (38) to generally provide different lim-its at the different sides of the BPs, as discussedabove for the case of the plane boundaries.

It should be also noted that there is one moretype of plausible singularities not yet discussed.These singularities have to appear if a given con-figuration has null points of magnetic field and thecorresponding separatrix field lines. In this case,the elemental flux tubes enclosing the separatrixfield lines split at each of the null points to pro-duce singularities in the derivatives of the Jaco-bian matrix. Such singularities are due to volu-metric rather than surface properties of magneticconfigurations. Therefore, if present, they appearin both Q and Q⊥ distributions thereby indicat-ing the existence of magnetic nulls in the corona.Since the numerical derivatives are estimated as aratio of finite coordinate differences, their absolutevalues may not exceed the ratio of the coordinaterange at the target boundary to a chosen incre-ment of coordinates at the launch boundary. Thepresence of this upper bound on possible values ofthe numerical derivatives prevents an overflow er-ror in computations of the squashing factors at thefootpoints of the null-point separatrices. A moredetailed consideration of this type of singularityrequires a special study, which goes far beyondthe scope of the present work.

4. COMPARISON OF Q and Q⊥

To see the difference between Q and Q⊥, theirdistributions are compared below for several mag-netic configurations. An emphasis is made on theirpotentiality for determining so-called hyperbolicflux tubes (HFTs). They are defined as two in-tersecting QSLs with extended and narrow pho-tospheric footprints characterized by very largevalues of Q. Neglecting the possible curving andtwisting of an HFT, its cross section variation inthe longitudinal direction can be respresented as

(42)

In other words, the width of one of the QSLsis shrunken to the thickness of the other in theprocess of the mapping of their cross sectionsalong magnetic field lines. This is possible be-cause the field lines in HFTs exponentially con-verge in one transversal direction and diverge in

the other. Such a property is typical for hyper-bolic flows in the theory of dynamical systems(Arnold 1988), which was coined in the term HFT(Titov et al. 2002). The examples below demon-strate that Q⊥ is a more accurate quantity thanQ for characterizing HFTs, although Q also pro-vides valuable complementary information on themagnetic structure.

4.1. The simplest possible hyperbolic flux

tube

The most simple magnetic configuration inwhich one would expect the presence of an HFTis the so-called X-line configuration, whose fieldB is determined in Cartesian coordinates (x, y, z)by

B = (−hx, hy, 1) = ∇ξ×∇η, (43)

where h = hL/B‖ is a dimensionless field gradi-ent characterizing the strength of the transversefield hL compared to the longitudinal field B‖ ona characteristic length scale L. The right-handside of (43) represents the same field in terms ofthe Euler potentials

ξ = x ehz, (44)

η = y e−hz, (45)

which are constant along the field lines. The useof ξ and η significantly simplifies the calculation ofthe squashing factors in this configuration. Sup-pose that its volume is restricted by |z| ≤ 1, sothat the planes z = ±1 are the correspondingboundaries. The constancy of ξ and η means thatthe boundary points (x−, y−) and (x+, y+) are re-lated by x−e

−h = x+eh and y−e

h = y+e−h. In

terms of the notations used in expressions (18) forthe plane boundaries, this means that X1 = e2hx1

and X2 = e−2hx2. Since Bz = 1 everywhere,equation (19) reduces simply to |∆| = 1, so thatthe resulting squashing factor is

Q = 2 cosh(4h). (46)

Thus, Q is constant over the entire planes z = ±1and, hence, it does not determine any QSL in theX-line configuration with the plane boundaries.Priest & Demoulin (1995) has arrived at the sameconclusion by using the norm N defined by (18),since Bz = 1, |∆| = 1 and so N =

√Q in this

8

configuration. However, they have found with thehelp of N some evidences of the presence of QSLsin the X-line configuration, if it is bounded by cu-bic, hemispheroidal and spherical surfaces.

On the other hand, it is clear from the gen-eral point of view that there should be an HFTin such a configuration irrespective of the shapeof the boundary surfaces. Indeed, this configu-ration has in the unbounded space two genuineseparatrix planes x = 0 and y = 0, which can beregarded as a limiting case of the bounded config-uration with the boundaries moved off to infinity.Taking into account also that h ∼ L, one couldexpect that the proper measure for QSLs must begrowing with h near the planes x = 0 and y = 0much stronger than in the remaining volume to in-dicate the corresponding QSLs near these planes.This would provide in the limit of large h an ex-pected continuous transition of such bounded con-figurations with QSLs to the unbounded config-uration with the genuine separatrices. The fail-ure of Q in determining an HFT in this simplecase looks very surprising in light of its remark-able success in other more complicated field con-figurations (see §4.2). The reason for this failurelies actually in the above-mentioned projection ef-fect, which is extremely large for the chosen typeof boundaries. The transverse component heregrows linearly with the distance from the X-linex = y = 0, while the longitudinal component re-mains constant. So the farther a given field linemeets the boundary from the X-line, the more theybecome aligned with each other.

Such an explanation is fully confirmed by calcu-lations of the perpendicular squashing factor Q⊥

in this case. To derive an analytical expression ofQ⊥, let us choose (ξ, η) as coordinates on bothboundary surfaces, so that u1 = w1 = ξ andu2 = w2 = η in (30)–(36). Assume for generalitythat the boundaries are defined by z = Z±(ξ, η),then according to (44) and (45) the vector func-tions

R(ξ, η) = (ξ e−hZ− , η ehZ− , Z−), (47)

R(ξ, η) = (ξ e−hZ+ , η ehZ+ , Z+) (48)

define, respectively, the launch and target bound-ary surfaces. These formulas are needed for calcu-lating (2), (3), (8), (31) (33), (34), (36) and (43)in the chosen coordinates (ξ, η). With the help of

such calculations, equation (30) yields

Q⊥ =2 cosh [2h(Z+ − Z−)] + h2

(

X2+ + Y 2

+ +X2− + Y 2

−

)

√

[

1 + h2(

X2+ + Y 2

+

)] [

1 + h2(

X2− + Y 2

−

)]

, (49)

in which

X± = ξ e−hZ± , (50)

Y± = η ehZ± (51)

determine (x, y) coordinates of the conjugate foot-points at the defined boundaries.

These expressions determine Q⊥ as a functionof the Euler potentials ξ and η. Note also thatξ = x and η = y at z = 0, so expressions (49)–(51) define in addition the distribution of Q⊥ inthe plane z = 0, to which Q⊥ is mapped along thefield lines from the boundaries. More generally,the combination of these expressions with equa-tions (44) and (45), resolved with respect to x andy as

x = ξ e−hz, (52)

y = η ehz, (53)

provides a parametrical representation ofQ⊥, withξ and η as parameters, in any plane z = constbetween the boundaries.

In the particular case of plane boundaries, wehave to put Z±(ξ, η) = ±1 in (49)–(51). Usingthen (44), (45), (50) and (51), expression (49) forQ⊥ can be rewritten even as an explicit function of(x, y, z). By comparing (49) and (46) one can seealso that the terms containing X± and Y± are re-sponsible in this case for eliminating the projectioneffect. As a result of this, the distribution of Q⊥

in the z = 0 plane shows very pronounced “ridges”along the x and y axes (Fig. 2a) by revealing anexpected HFT with a characteristic X-type inter-section of QSLs along the X-line. According toequations (52) and (53), such a structure shrinksand expands along these axes exponentially fastwith z to give a typical HFT variation of its crosssection (see diagram [42]). With growing h, theridges of the Q⊥ distribution and so the corre-sponding QSLs become thinner and thinner byextending on larger and larger distances from theX-line. Thus, the perpendicular squashing factorQ⊥ defines indeed an HFT such that it continu-ously transforms in the limit of large h into theseparatrix planes x = 0 and y = 0.

9

Fig. 2.— Distributions of logQ⊥ in the plane z = 0 for the X-line configuration from eq. (43) at h = 2.The boundaries are two parallel planes z = ±1 (a) or two hyperbolic paraboloids h(x2 − y2)/2− z = ∓1 (b).In the second case, Q⊥ = Q, since the field lines are strictly orthogonal to the paraboloids. The contoursin both distributions correspond to Q⊥ = 100. In the second case, such a contour represents also the crosssection of the HFT by the plane z = 0, so that the field lines passing through this contour form the lateralsurface of the HFT (c) bounded from the top and bottom by the hyperbolic paraboloids.

10

However, the peripheral field lines in this HFTare still nearly parallel to the boundaries z = ±1,which causes too strong of a variation of the HFTcross section. Aesthetically more pleasing HFT inthe X-line configuration can be obtained by usingthe boundaries which are orthogonal to the fieldlines. These are iso-surfaces of magnetic potentialF = h

(

x2 − y2)

/2 − z having the shape of hy-perbolic paraboloids. It is natural to chose thempassing through the points (x = y = 0, z = ±1),which means that F = ∓1 for such iso-surfaces.This condition using (50) and (51) yields

h2

2

(

ξ2e−2hZ± − η2e2hZ±)

− Z± ± 1 = 0, (54)

which is a transcendental equation for the Z±

functions entering in expression (49). This equa-tion is not difficult to solve numerically for givenξ and η and to use the respective solution for thecalculation ofQ⊥. An example of the resultingQ⊥

distribution in the plane z = 0 at h = 2 is shown inFigure 2b. The corresponding HFT with the mag-netic surface defined by Q⊥ = 100 and boundariesF = ∓1 is presented in Figure 2c. For the chosentype of boundaries, Q⊥ = Q, so both squashingfactors define the same HFT. The magnetic fieldfor this HFT has the simplest analytical form; thehyperbolic paraboloids are also relatively simpleboundary surfaces. So we believe that this exam-ple provides the simplest possible HFT relevant fortheoretical studies of basic MHD processes, suchas magnetic pinching and reconnection, in threedimensions.

4.2. HFT in twisted magnetic configura-

tion

The considered X-line configuration with z =±1 boundaries is an important example, whereQ⊥, in contrast to Q, succeeds in determining anexpected HFT. However, this example is not rep-resentative enough to make a general conclusionon the potentialities of Q and Q⊥ for detectingQSLs. Because the field lines in such a configura-tion behave in a rather artificial way over a majorpart of the boundaries. A better comparison ofQ and Q⊥ can be done by using a more realisticfield.

For this purpose, we have chosen the analyticalmodel of a twisted magnetic field (Titov & Demoulin

1999), hereafter called the T&D model. It de-scribes approximate equilibria of a circular mag-netic flux rope, whose interior force-free field iscontinuously embedded into a potential back-ground field. The latter is produced by fictitioussubphotospheric sources consisting of two mag-netic monopoles of opposite signs and a line cur-rent, all located at the axis of symmetry of therope. The axis itself is placed some depth belowthe photospheric plane and the minor radius ofthe rope a is assumed to be much smaller thanthe major one Rc and the distance between themonopoles L.

To compare Q and Q⊥ in detail, we have com-puted their distributions for three sets of parame-ters which differ only in values of Rc. Two of thesevalues (Rc = 85 and 98Mm) and all the remainingparameters are chosen to be exactly the same asin the T&D model. By growing Rc but keepingother parameters fixed we imitate an emergenceof the flux rope from below the photosphere. Inthis process, the configuration passes continuouslythrough three distinct topological phases. For suf-ficiently small Rc, there is a single BP separatrixsurface (Titov & Demoulin 1999)—the configura-tion with Rc = 85 Mm represents one of thesetopological states. The corresponding distribu-tions of Q and Q⊥ are shown in Figures 3a and3d, respectively. With growing Rc, this BP andthe associated separatrix surface bifurcate into twoparts to give birth to a BP separator field line(Titov & Demoulin 1999)—the configuration withRc = 98 Mm represents this second topologicalphase. The corresponding distributions of Q andQ⊥ are shown in Figures 3b and 3e, respectively.The points Sa and Sd on these figures are the foot-points of the BP separator, while Sb and Sc are itsphotospheric contact points.

Further growing of Rc leads to a complete dis-appearance of the bifurcated BP and the associ-ated separatrix surfaces—the configuration withRc = 110Mm represents this last phase. The cor-responding distributions ofQ andQ⊥ are shown inFigures 3c and 3f, respectively. The magnetic fieldat these parameters becomes topologically trivial,since its field line mapping is continuous every-where. The whole structure can be continuouslytransformed to a simple arcade-like configurationwith the help of a suitable photospheric motion.Thus, with growingRc or emerging of the flux rope

11

Fig. 3.— Distributions of logQ ([a], [b] and [c]) vs. logQ⊥ ([d ], [e] and [f ]) at different values of the majorradius Rc of the flux rope and with other parameters of the model fixed. The contours of the photosphericnormal magnetic field Bz = 0,±100,±200 and ±400 G (and additionally Bz = ±300 G for [c] and [f ]), theBPs (thick solid segments of the PIL), and footprints of the BP separatrices (dotted thin lines on [a], [b], [d]and [e]) are also shown. The footprints Sa and Sd and touch points Sb and Sc of the BP separator field lineare indicated on panels (b) and (e).

12

Fig. 4.— Same as Fig. 3, but with the shading saturated at the level 2.5 to reveal the difference betweenthe distributions at moderate values.

13

the topological complexity of the configurationfirst increases and then abruptly decreases.

On the contrary, the geometrical distortion ofthe field line mapping gradually increases in theconfiguration during this process. Both squashingfactors Q and Q⊥ continue to grow with Rc in nar-row strips of the photospheric plane. It is clearlyseen from the Q and Q⊥ distributions plotted withthe same grayscaling (Fig. 3) that Q > Q⊥ ev-erywhere and the difference between Q and Q⊥

becomes smaller with growing Rc. The most sig-nificant difference between them is seen at the firsttwo phases near the BPs and the footprints of theassociated separatrix surfaces. As previously an-ticipated, Q always rises in these regions, whileQ⊥ does not, except near the contact points Sband Sc of the BP separator (Fig. 3e). The valueQ⊥ does rise there but only to give birth to a partof HFT footprints, which are matured eventuallyin the third phase (Fig. 3f). As concerned with theindicated rise of Q, it is mainly caused by the pro-jection effect: the field lines which are close to theBP separatrix surfaces approach the photospherenear the BPs at a small angle to the horizontal,which strongly distorts the footprints of the corre-sponding elemental flux tubes. Comparing Q andQ⊥ at Rc = 110Mm reveals that the latter is validfor the central part of the PIL as well.

The discussed features of the distributions be-come more transparent if one saturates the greyshading in the plots at the values ≥ 2.5 (Fig. 4).These new plots show that the QSLs based onthe Q⊥ distribution are characterized by a thin-ner and more uniform thickness. Their footprintsacquire atRc = 110Mm a clear fishhook-like shapein each of the photospheric polarities with maxi-mums of Q reaching ∼ 108. The QSLs rooted atsuch “fishhooks” intersect each other by combin-ing themselves into an HFT (Titov et al. 2003).This structural feature seems to be very robust,because it appears even in twisted configurationswhich are not in force-free or magnetostatic equi-librium (Demoulin et al. 1996b). One can see fromFigure 5a that, except for an essential twisting dis-tortion, the cross section of such an HFT variesexactly according to diagram (42).

It has yet to be proved, but it seems to bequite natural that this HFT is pinched into a ver-tical current sheet below the flux rope by its up-ward movement when the kink or torus instabil-

ity is developed in the configuration at a suffi-ciently large twist of the field lines in the rope(Torok et al. 2004; Roussev et al. 2003). Thisinterpretation is very important for understand-ing the properties of sigmoidal structures in flar-ing configurations (Kliem et al. 2004). Figure 5asuggests that the sigmoids in such configurationsare simply pinching HFTs illuminated by a hotplasma material which appears there due to thereconnection process in the above-mentioned ver-tical current sheet. This seems to be valid atleast for the third topological phase of the fluxrope emergence, while an additional interactionwith the photosphere must be involved at the firstand second topological phases, where the BPs arepresent (Titov & Demoulin 1999; Fan & Gibson2003). Panels (d)—(f) in Figures 3 and 4 demon-strate that the footprints of the BP separatrixsurfaces follow very close to the HFT footprintsemerging gradually with growing Rc. This impliesthe corresponding similarity in the shapes of suchseparatrix surfaces and HFTs. So the explanationsof the sigmoids that rely on either the presence ofthe BPs (Titov & Demoulin 1999; Fan & Gibson2003) or the HFTs (Kliem et al. 2004) are not al-ternative but rather complementary, since they re-fer to different phases of the flux rope emergence.

It should be noted also that both Q and Q⊥

distributions (Fig. 4) contain at the border of theflux rope two less pronounced horseshoe-like fea-tures with maximums ∼ 102. The QSL rooted atthese “horseshoes” has a helical shape (Fig. 5b)with a slightly varying cross section along the fieldlines. So this QSL has a structure qualitatively dif-ferent from those two which form the HFT. Thecomparison of Q and Q⊥ shows that the squash-ing of the flux tubes in this helical QSL is only inpart due to the projection effect. The Q⊥ distri-bution demonstrates that the major contributionto Q comes from the shearing of the twisted fieldlines in the rope. Thus, both distributions reveala helical QSL which is a part of the inner borderlayer of the flux rope. In this respect, the con-sidered example demonstrates that our squashingfactors help identify flux ropes themselves. It isnot a problem, of course, to locate the flux rope inthe T&D model, where its parameters are knownfrom the construction of the model. Yet identify-ing flux ropes in more complicated configurationsobtained, for instance, numerically from magne-

14

Fig. 5.— HFT (a), the helical QSL (c) and their cross sections, for the HFT (b) and for the QSL (d), inthe T&D twisted magnetic configuration at Rc = 110 Mm. Other parameters of the model, as well as thecontours of the photospheric normal magnetic field Bz, are the same as in Figs. 3c and 3f; the white stripeson the photosphere represent the HFT and QSL footprints, whose contours correspond to Q = 100.

15

togram data is a real problem. If such configu-rations are topologically trivial, like those in thethird phase of our example, the determination ofQSLs seems to be the only method for identifyingflux ropes.

As shown above, both squashing factors allowus in the case of the T&D model to determinesimilar QSLs, except that Q⊥ is more advancedthan Q near BPs, whenever they appear. There-fore, we think that, in general, if the numericalgrid used for computing Q⊥ is fine enough for de-tecting possible BP separatrix surfaces by suddenspikes in the Q⊥ distribution, the use of only Q⊥

would be sufficient for the structural analysis ofconfigurations. In practice, however, the requiredresolution of the grid cannot be always easily fore-seen. Also the computational cost for Q is notreally high, since the same input data as for Q⊥

can be used. So it is sensible to compute boththese distributions at a time and compare themin the same way as we did in the considered ex-ample. Some redundancy of the information con-tained in such distributions is not superfluous butuseful, especially, in the case of complicated realmagnetic configurations. Thus, from the practicalpoint of view, the value Q should not be consid-ered as obsolete but rather as a complementarycharacteristic of magnetic connectivity.

5. SUMMARY

We have derived a covariant form of the squash-ing factor Q, which enables us to determine quasi-separatrix layers (QSLs) in both closed and openmagnetic configurations with an arbitrary shape ofboundaries. The corresponding expression for Qassumes that the Jacobian matrix of the field linemapping and the metric tensors at the footpointsare known. The expression admits also that such“input data” can be represented with the help oftwo different coordinate systems for determininglocation of the conjugate footpoints on the bound-aries. This provides a firm theoretical basis for aglobal description of the field line connectivity inthe solar corona.

To eliminate the projection effect at the fieldlines which are nearly touching the boundary, theperpendicular squashing factor Q⊥ is also derivedin a similar covariant form. The value Q⊥ definesthe degree of squashing of elemental magnetic flux

tubes only in the directions orthogonal to the fieldlines. In the definition of Q⊥, the boundaries en-closing the magnetic configuration constrain onlythe length of the flux tubes while not affectingtheir cross sections at the footpoints. For calcu-latingQ⊥, the vectors of magnetic field at the foot-points are required in addition to the same inputdata as for Q. The use of both covariant squash-ing factors is demonstrated by calculating themfor the boundaries with the planar and sphericalgeometries. Then the properties of Q and Q⊥ arecompared by considering two examples of mag-netic configurations.

The first example is a classical X-line configura-tion of potential magnetic field in a plasma volumerestricted by two boundary surfaces. It is easy toshow that for the plane boundaries perpendicularto the X-line, the value Q is constant. So the Qdistribution does not allow us to define any QSLsin such a configuration. The reason for this failurelies in the projection effect, which is very strongfor the field lines distant from the X-line. We havealso calculated an analytical expression of Q⊥ forthe same field but with the boundaries of an ar-bitrary shape. In the case of the plane bound-aries, this new value Q⊥, in contrast to Q, hasa non-uniform distribution, which does reveal theexpected two QSLs. These QSLs intersect eachother by combining themselves into a hyperbolicflux tube (HFT). A more elegant HFT is obtainedfor the X-line configuration with the boundariesorthogonal to the field lines.

To make a better comparison of the proper-ties of Q and Q⊥, a second magnetic configura-tion more relevant for solar physics is considered.The respective field is defined by using the Titov-Demoulin (1999) model of a force-free flux ropeembedded into a potential background field. Con-trary to the case of the X-line configuration re-stricted by the plane boundaries, both the Q andQ⊥ distributions reveal QSLs in the twisted con-figuration. These distributions are similar every-where except near bald patches (BPs) and foot-prints of the associated separatrix surfaces, when-ever the BPs exist. By definition, the value Q⊥

is free of the projection effect, so that Q⊥ risesnear BPs only if the corresponding flux tubes aresubject to a volumetric squashing. This is not thecase, of course, for the value Q, which always risesin such regions of the photosphere. So, in compar-

16

ison with Q, the value Q⊥ shows itself to be againa superior characteristic for analysis of magneticconnectivity.

Nevertheless, we have argued that it is morepractical in general to compute both squashingfactors for analyzing the structure of a given mag-netic configuration. This does not require addi-tional significant effort, while making it easy todiscriminate between the volumetric squashing ofelemental flux tubes and the surface projection ef-fect at the boundaries.

This research was supported by NASA and theCenter for Integrated Space Weather Modeling (anNSF Science and Technology Center).

REFERENCES

Arnold, V. I. 1988, Geometrical methods in thetheory of ordinary differential equations (NewYork, Springer-Verlag, 364 p.)

Baum, P. J., & Bratenahl, A. 1980, Sol. Phys., 67,245

Demoulin, P., Bagala, L. G., Mandrini, C. H.,Henoux, J. C., & Rovira, M. G. 1997, A&A,325, 305

Demoulin, P., Henoux, J. C., Priest, E. R., & Man-drini, C. H. 1996a, A&A, 308, 643

Demoulin, P., Priest, E. R., & Lonie, D. P. 1996b,J. Geophys. Res., 101, 7631

Fan, Y., & Gibson, S. E. 2003, ApJ, 589, L105

Kageyama, A., & Sato, T. 2004, Geochemistry,Geophysics, Geosystems, 5, 9005

Kliem, B., Titov, V. S., & Torok, T. 2004, A&A,413, L23

Lau, Y.-T., & Finn, J. M. 1990, ApJ, 350, 672

Levine, R. H., Schulz, M., & Frazier, E. N. 1982,Sol. Phys., 77, 363

Longcope, D. W. 2001, Physics of Plasmas, 8, 5277

Longcope, D. W., & Cowley, S. C. 1996, Physicsof Plasmas, 3, 2885

Longcope, D. W., & Strauss, H. R. 1994a, ApJ,426, 742

—. 1994b, ApJ, 437, 851

Marsden, J. E., Ratiu, T., & Abraham, R. 2002,Manifolds, Tensor Analysis, and Applications,3rd edn., Applied Mathematical Sciences (NewYork, Springer-Verlag, 555 p.)

Parker, E. N. 1979, Cosmical magnetic fields:Their origin and their activity (Oxford, Claren-don Press; New York, Oxford University Press,858 p.)

—. 1994, Spontaneous current sheets in magneticfields: with applications to stellar x-rays (NewYork, Oxford University Press)

Priest, E. R., & Demoulin, P. 1995, J. Geo-phys. Res., 100, 23443

Priest, E. R., & Forbes, T. G. 2000, MagneticReconnection: MHD theory and applications(Cambridge, UK: Cambridge University Press,612 p.)

Priest, E. R., & Titov, V. S. 1996, Philos. Trans.R. Soc. London A, 354, 2951

Roussev, I. I., Forbes, T. G., Gombosi, T. I.,Sokolov, I. V., DeZeeuw, D. L., & Birn, J. 2003,ApJ, 588, L45

Sweet, P. A. 1969, ARA&A, 7, 149

Syrovatskii, S. I. 1981, ARA&A, 19, 163

Titov, V. S., & Demoulin, P. 1999, A&A, 351, 707

Titov, V. S., Demoulin, P., & Hornig, G. 1999, inESA SP-448: Magnetic Fields and Solar Pro-cesses, ed. A. Wilson & et al., 715–722

Titov, V. S., Demoulin, P., & Hornig, G. 2003,Astronomische Nachrichten, 324, 17

Titov, V. S., & Hornig, G. 2002, Advances inSpace Research, 29, 1087

Titov, V. S., Hornig, G., & Demoulin, P. 2002,J. Geophys. Res., 107, 3

Titov, V. S., Priest, E. R., & Demoulin, P. 1993,A&A, 276, 564

Torok, T., Kliem, B., & Titov, V. S. 2004, A&A,413, L27

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