Time=1.2. Time=0. Time=1.5. Time=1.1

Gunnar Hornig & Lutz Rastatter

Theoretische Physik IV, Ruhr-Universitat, D-44780 Bochum, Germany

December 8, 1996

Abstract

Magnetic reconnection can be interpreted as a process in which the electromag-netic eld is frozen into a four-velocity eld in Minkowski space. For reconnec-tion to occur the four-velocity eld has to be a special type of stagnareconnec-tion ow. Prescribing this type of ow in a nite spatial domain allows the modelling of localized reconnection events and the investigation of examples of reconnection in regions without magnetic nulls. In the present contribution, we start with a simple twisted magnetic ux tube. Reconnection occuring along a part of the axis of the tube results in a structure of the magnetic eld which is a superpo-sition of a two-dimensional X-type magnetic eld well-known from stationary 2D reconnection models, and a component resulting from the magnetic eld parallel to the axis. For localized reconnection, the latter component of the magnetic eld evolves in a non-trivial way. This evolution is important for the spatial variation of the parallel electric eld integrated along the magnetic eld lines. The integrated electric eld gives an upper limit for the energy to which particles can be accelerated in a reconnection event and its distribution shows to be localized in very thin structures.

1 Introduction

The basic notion of magnetic reconnection was formed by two-dimensional models, which all have in common that reconnec-tion takes place at magnetic X-points (neutral lines in three di-mensions, respectively). Although some kind of non-idealness is necessary for reconnection to occur, these models satisfy, as in ideal dynamics,

E

B

= 0 due to their simple geometry. A

foundation of reconnection for

B

6= 0 and accordingly

E

B

6= 0

was given by Hesse and Schindler [1, 2]. This latter form of reconnection is more realistic in the sense that a

B

= 0-line is a structurally unstable conguration and thus in a realistic case at least a small component of the magnetic eld parallel to reconnection line will exist. Such a conguration is found for instance in the magnetotail of the Earth for reconnection at a near earth x-line, which usually has a parallel component of the magnetic eld due to the distortion of the magnetotail by the solar wind. Another example is the reconnection in the lower magnetosphere which may accelerate electrons along the magnetic eld into the ionosphere and thus cause aurorals arcs [3]. In both examples the structure of the magnetic eld plays a crucial role for the interpretation of observations. This is our motivation to model a 3-D reconnection event localized in space and time. Because we are primarily interested in the structure of magnetic and electric elds in the process of reconnection, we restrict ourselves to a kinematic model. For reasons which are explained below we use a covariant set of equations which is solved numerically with a method described in an accompa-nying paper [4].

2 B

= 0

versus B

6= 0

reconnection

The two-dimensional reconnection occurs at an X-point of the magnetic eld which in three dimensions corresponds to a mag-netic neutral line. These models usually have a simple geome-try such that

E

B

= 0 holds everywhere. A very simple

exam-ple conguration for two-dimensional stationary reconnection in Cartesian geometry for a region 2x2+y2<1 is given by

E

+

v



B

=

J

(1) with

E

= 1

e

z

B

= y

e

x+ 2 x

e

y (2)

v

=?x

e

x+ y

e

y

J

= 1 =0

e

z  =0(1?y 2 ?2x 2₎ :

Due to the non-ideal term 

J

the magnetic eld is not

every-where frozen into the plasma velocity

v

in this example. As we will show, due to

E

B

= 0 we can nd another velocity eld

w

into which the magnetic eld is frozen. From a mathematical point of view, the magnetic ux is frozen into a velocity eld

w

if a potential  exists such that

E

+

w



B

=r (3)

holds. The magnetic ux through a surface comoving with the velocity

w

is conserved. With the help of the transporting velocity eld

w

, we can actually interpret reconnection as the typical "cutting" and "pasting" process of magnetic ux which it is often considered to be. For the example stated above (as in every case with

E

B

= 0), we can actually nd a velocity

eld

w

satisfying (3) with  = 0 which here has the form

w

:= (1 +J z

=(2x2+y2))

v

This velocity eld has the same stagnation ow features as the plasma velocity, but is singular on the z-axis. This singularity of the transporting ow is crucial for the reconnection of mag-netic ux at the stagnation point, because the magmag-netic eld is transported towards the z-axis in a nite amount of time. At the z-axis it splits and is reconnected to form a new ux tube, which is then transported outwards again. Note that this cannot be achieved by any continuous velocity elds, be-cause for these ows the time necessary for a uid element to reach the stagnation point is always innite. For magnetic ux conserving ows this results in a pile-up of ux at the stagna-tion point. A sketch of both a continuous and a singular ow situation is shown in Figures 1 and 2 for a simple magnetic ux ring. In Figure 1 the ux ring is frozen into a continuous stagnation ow resulting in a ux pile-up. In Figure 2 a non-continuous transporting velocity of the above mentioned type yields reconnection.

Time=0

W W _Time=0.7

Time=1.1

W

Time=1.5

Figure 1: Pile-up of a magnetic ux tube in a stagnation ow

Time=0 Time=1.2

Time=1.7 Time=2.3

Figure 2: Reconnection of a magnetic ux tube in a singular stagnation ow

We were able to derive a transporting velocity

w

in the pre-ceding example because for

E

B

= 0 one can always dene

w

:= (

E



B

)=

B

2

: (4)

This velocity eld may have singularities at null-points of the magnetic eld and if these are of the type mentioned above, i.e. a stagnation ow which allows for a transport onto the stagnation point in a nite time, it will lead to reconnection. If the non-ideal term (

J

in our example) is conned to a certain

region outside of which the ideal form of Ohm's law holds, then the transport velocity

w

is equal to the plasma velocity in the ideal region up to a component parallel to

B

. We can use this transporting ow to dene reconnection as a magnetic ux conserving process having a stagnation-type transporting ow with a singularity along a certain line, the reconnection line. This singularity has to allow for a transport onto the stagnation point in a nite time, i.e. the time necessary for a particle, starting at a point

x

on the trajectory which ends at the stagnation point (

x

0), to reach

x

0 is nite.

Z x0 x dt= Z x0 x 1 k

w

k dx<1: (5)

This meets all forms of reconnection with

E

B

= 0. For

B

6= 0 at the reconnection site, i.e.

E

B

6= 0, our denition

does not apply because these cases are usually not magnetic

ux conserving. If they would be magnetic ux conserving a denition of a transporting velocity analogous to (4),

w

:= (

E

?r)

B

)=

B

2

leads (for smooth elds

E

;

B

6= 0 and ) always to smooth

w

which leads to the contradiction that we could regard the magnetic eld as smoothly transported with no reconnection taking place. We get an example for a B 6= 0-reconnection if

we modify the example from above by adding a constant B z

-component and replace the electric eld by

E

= (yB z

;xB z

;1).

From the kinematic point of view the parallel component of the magnetic eld is in this case only superposed to the recon-necting perpendicular components and plays no active role in reconnection process. To include this case in a our denition of reconnection we have to note that magnetic ux conserva-tion is only a special case of a more general conservaconserva-tion of electromagnetic ux [5],[6] which uses the relativistic covariant equations

?

E

W

=@0 (6)

W

0

_E

₊

_W



B

=r: (7)

where the index 0 denotes the time coordinateX0=ct. These

equations imply the conservation of electromagnetic ux

Z C

B

d

A

+ Z C

E

d

x

dx0=const: (8)

for a surface comoving in four-dimensional space (Minkowski space) with the four-velocityW(4)= (W0;

W

). The rst term

in eq. (8) is the usual magnetic ux penetrating the surface (C), while the second term is due to a possible extension of the surface along the time axis. For W0 1 we recover the

magnetic ux conserving evolution. For W0 = W0(

x

;t) we

have some kind of non- ux conserving and therefore non-ideal evolution. We can regard the four-velocity W(4) as a

trans-porting velocity of the electromagnetic ux. This transtrans-porting ow can be prescribed in such a way that reconnection occurs. First we can reproduce the

E

B

= 0-reconnection from above

with the help of  = 0 and a four-velocity W(4) where the

space components resemble a two-dimensional stagnation ow, i. e.

W

= (?x;y;0) and a time component W0 = 2x2+y2.

Note that the four-velocity in the Minkowski space is perfectly smooth, but the projection

W

(4)

!

w

=

W

=W

0 ₍₉₎

has the singularity necessary to produce reconnection. In this more general ux conservation, reconnection for

E

B

6= 0 is

not excluded any more. The

B

6= 0 example from above for

instance is electromagnetic ux conserving for a transporting velocity W (4) _{= 1} =(x 2_{+ 2} y 2_{+ 1)[2} x 2₊ y 2 ;?x;y;0]  = xy

The existence of such a transporting velocityW(4) can be used

for a denition of reconnection by replacing in the above de-nition magnetic ux conservation by electromagnetic ux con-servation and

w

by

W

=W0. Note that W0 as well as the

space components

W

are for

B

6= 0 of the same type near

the z-axis as in the

B

= 0 case. To give an impression what electromagnetic ux conservation means for reconnection as-sume that we have a nonvanishing magnetic ux penetrating

a surface located in the x-z-plane. This surface is transported onto the z-axis with the velocityW

x= dX ds =

?xparametrized

by s. Hence, X(s) =X(0)exp(?s) and the surface shrinks in

x-direction as it approaches the z-axis for s!1(see Figure

3). Now the relation between our parameter s and the time is given byW0 andW0= dt ds leads to t(s) = Z W 0₍

_x

;t)ds= Z W 0 =W x dx:

The function W0 determines whether the time in which the

s=0 s=0.4 s=1 s=3 0 1 2 3 4 5 x 0 0.5 1 1.5 2 z 0 2 4 6 8 t

Figure 3: During the transport of a surface along the x-axis onto the reconnection line (z-axis) the magnetic uxR

C

B

d

A

is converted intoR C

E

d

z

dt.

surface reaches the z-axis is nite or not. For our example the choice of W0 = x2+y2 yields a nite time while for all

functions where W0 > 0 at the stagnation point the time is

innite. For the latter case a ux tube never reaches the stag-nation point. For the case of nite time the magnetic ux is completely converted into electric ux when the surface has reached the z-axis. This can be deduced from the conservation of electromagnetic ux (eq. (8)) where the contribution of the magnetic ux vanishes at the stagnation point. Therefore the reconnected ux is given by

Z C

E

d

z

dt= Z C

B

d

A

: (10)

Along the out ow direction the process is exactly the other way round, the electric ux is converted into magnetic ux again. Hence, if we assume thatW0diers from 1 only in a localized

region and if we do not resolve this non-ideal region, i.e. if we look at it on a large scale, the whole process regarding the magnetic ux perpendicular to the reconnection line (z-axis) looks quite the same as in the ideal case. And indeed due to the linearity of the equations (6),(7) we can separate the

E

B

6= 0-reconnection into an ideal and a non-ideal part. 

E

=

E

id+

E

nid

B

=

B

id+

B

nid (11)  ?

E

id

W

= 0 W0

E

id+

W



B

id= 0 (12)  ?

E

nid

W

=@0 W0

E

nid+

W



B

nid= r (13) This decomposition is generally not unique, because we can always add a part of the ideal evolution to the non-ideal evo-lution. However, we can use this freedom to set

E

nid

k = 0 at

the reconnection line dened by the vanishing ofW4. The

sub-scripts and are used with respect to the reconnection line.

This line is tangential to an eigenvector (eigenvalue 0) ofr

W

atW0= 0 and forms together with the other two eigenvectors

of r

W

(positive and negative eigenvalue) a local coordinate

system. It is possible to dene

E

nid

k = 0 because from (7) we deduce W 0

_E

B

=

B

r (14) which, requires r

k = 0 at the reconnection site due to the

vanishing of W0 while

B

6= 0 and

E

B

6= 0. This yields a

unique decomposition at least for a neighborhood of the recon-nection line and means that the non-ideal magnetic eld does not contribute to the reconnection (cf. (10)).

The decomposition then requires the following structure of parallel and perpendicular components in the lowest nonvan-ishing order

B

id k = 0 + O(x) (15)

E

id ? = 0 + O(x)

E

id k = ?

W



B

id ? =W 0 6 = 0 (16)

B

nid ? = 0 + O(x)

B

nid k = const:6= 0 r k = 0 + O(x 2₎ r ? =

W



B

nid k + O(x 3₎

E

nid k = 0 + O(x)

E

nid ? =  ?

W



B

nid k + r ?  =W 0

In our standard example the ideal components are those given in (2) the non-ideal components are

E

nid:= 0,

B

nid:= (0

;0;1)

and  =xy, corresponding to a constantB

z-component.

3 Reconnection and magnetic line conservation

Another important aspect which distinguishes

E

B

= 0 from

E

B

6= 0-reconnection is the possibility to dene a

trans-porting velocity of the magnetic eld lines. For

E

B

= 0

the electromagnetic ux conservation reduces to magnetic ux conservation and this implies the conservation of magnetic eld lines as well [6]. That is, we can map the transporting veloc-ity for the electromagnetic ux onto a transporting velocveloc-ity of the magnetic eld lines with the help of relation (9). Thus we can distinguish dierent eld lines during their evolution. (We used this property for instance to identify the ux tube shown in Figure 1 and 2.) For

E

B

6= 0 usually no such transporting

velocity exists. The mapping (9), for instance, yields a trans-porting velocity for the magnetic eld under

E

B

6= 0 only if

the induction equation, which we derive from (7)

@ t

B

+ r ?

W

=W 0 

B

=r(1=W 0₎ r

has the eld line conserving form

@ t

B

+ r ?

W

=W 0 

B

=

B

:

This implies rW0r = 0 because otherwise

B

r = 0

due to (14) would require

E

B

= 0. Another velocity dierent

from W=W0 may of course exist which may serve as a eld

line transporting ow. For instance any magnetic eld which can be obtained by a distortion of the homogeneous eld (e.g. a ux tube), and develops no nulls in the volume under consider-ation possesses a transporting velocity for the eld lines. (Such elds also have Euler-potentials [1].) But there is no guarantee for the existence of such a velocity eld in the general case. Moreover, we can nd examples which obviously have no such a transporting velocity for the eld lines. The appearance of new magnetic ux in the non-ideal region is an example where no transporting velocity for the eld lines can exist. Generally

the appearance or vanishing of magnetic nulls indicates that no dierentiable velocity eld exists which is magnetic eld line conserving [7]. An example for such a case is given in [8]. How-ever, if we use the splitting (11) we at least have a transport velocity for the ideal part of the electromagnetic eld.

4 3-D localized reconnection (E

B

= 0

)

According to our denition reconnection requires a transport-ing velocity W(4) which vanishes along a certain line and for

which the space components show a stagnation ow in the neighborhood of this line. This enables us to simulate recon-nection by prescribing this type of transporting four-velocity and use the equations (6),(7) to calculate the evolution of elec-tric and magnetic elds. The numerical method we use is ex-plained in [4]. To model a localized reconnection process we use a prole ofW0 which vanishes along a part of the z-axis.

This part of the z-axis is enclosed in a transition zone where

W0 smoothly increases to the value of 1. In the remaining

part of the volume W0 is constant (=1) (see Figure 4). The

space components ofW(4)have no z-component and the x and

y components form a stagnation ow of the type shown at the bottom of Figure 4 (left side). The strength of this ve-locity eld approaches zero at the boundaries of the volume which extends from -1 to 1 along each of the coordinate axes. Therefore,

W

=W0 is singular on the z-axis between -0.4 and

W -1 -0.5 0 0.5 1 x -1 -0.5 0 0.5 1 y 0 0.2 0.4 0.6 0.8 1 W^0 -1 -0.5 0 0.5 1 Y -1 -0.5 0 0.5 1 Z 0 0.2 0.4 0.6 0.8 1 W^0

Figure 4: Localization of W0 in the plane z = 0 (left) and

along the z-axis (right).

0.4, where W0 is zero, in a way which satises the condition

(5). Above and below of this part of the z-axis

W

=W0 is a

smooth stagnation ow which implies a pile-up of magnetic ux and no reconnection. The eect is shown in Figure 5 for three dierent ux tubes of a magnetic eld which is a circular eld having a maximum at a distance of 0.3 from the z-axis and approaches zero on all boundaries (the same is used for the starting conguration in Figure 1 and 2). For this cong-uration

E

B

= 0 holds everywhere and the four-velocityW(4)

corresponds to a transporting velocity

W

=W0, which acts on

the magnetic ux for eachz=const:-plane independently.

-0.5 0 0.5 x -0.5 0 0.5 y -1 -0.5 0 0.5 1 z -0.5 0 0.5 x -0.5 0 0.5 y -1 -0.5 0 0.5 1 z

Figure 5: Eect of localized reconnection forB z= 0.

5 Localized reconnection (E

B

6= 0

)

Now a constant B

z-component is superposed on the starting

conguration. The sum of both elds has a foliation of cylindric ux surfaces nested around the z-axis (see Figure 6, left side). They are distorted as the reconnection occurs in a way shown in

-0.5 0 0.5 x -0.5 ₀ 0.5 y -0.5 0 0.5 z -0.5 0 0.5 x -0.5 ₀ 0.5 y -0.5 0 0.5 z

Figure 6: Distortion of a ux tube due to a localized reconnec-tion process on its axis.

the right hand part of Figure 6, but contrary to theB

z= 0 case

the ux tube does not break up. This would require a magnetic null somewhere on the z-axis which we do not have due a nite z-component of the magnetic eld. TheB

z-component belongs

to the non-ideal part of the electromagnetic eld according to our decomposition (11) and therefore evolves independently of the ideal components. The corresponding non-ideal electric eld is given by

E

nid= ? ?

W



B

nid+ r
=W 0

The potential  is chosen in such a way that the numerator vanishes in the inner reconnection region whereW0approaches

zero. However,

E

nid cannot vanish everywhere because the

localization of the reconnection region in z-direction requires that  approaches zero at the boundaries. This implies a z-component of r and hence a z-component of

E

nid. This leads to an increasingB nid x and B nid y -components localized in

the transition regions above z = 0:4 and below of z = ?0:4

of the type shown in Figure 7. The corresponding form of

B

nid for the

x = 0-plane and y = 0-plane is shown in

Fig-ure 8. The vanishing of x and y-components of

E

nid imply -0.5 0 0.5 y -0.5 0_x 0.5 Figure 7: B nid x and B nid

y -components localized in the

transi-tion regions above z = 0:4 and below of z = ?0:4 (reversed

direction).

that the z-component of

B

nid is constant. Therefore no

mag-netic nulls appear during the reconnection and according to the last section there exists a dierentiable transporting ve-locity for the magnetic eld lines. However, this transporting

-0.5 0 0.5 z -0.5 0_x 0.5 -0.5 0 0.5 z -0.5 0y 0.5 Figure 8: B nid in the y= 0 andx= 0-plane. velocity is not

W

=W

0_{, which has a singularity, nor does it}

ap-proach

W

=W0outside the reconnection region whereW0= 1

and the magnetic eld has an ideal evolution. Thus the lo-calized reconnection has global eects on the magnetic eld topology (see also [1]). This eect of localized reconnection on the global topology can be studied with the help of the map-ping of points on the lower boundary along the eld lines to the upper boundary. This mapping is smooth due to the nite

B

z component, but it also shows steep gradients which are

characteristic for reconnection. It is therefore useful to detect reconnection as pointed out by Priest & Forbes in [9] and Priest & Demoulin [10]. Figure 9 shows the mapping of a circle at the lower boundary (z=?1) onto the upper boundary (z= 1). It

Time=1. -0.5 0 0.5 y -0.5 0 0.5 x Time=1. -0.5 0 0.5 y -0.5 0 0.5 x

Figure 9: The cross-sections atz=?1 andz= 1 for the nal

state of the ux tube shown in Figure 6, and for a ux tube with twice the twist.

shows a characteristic S-shape distortion which we found in all similar congurations. The extent of this distortion depends on the twist of the ux tube at the beginning. More windings of the eld lines around the z-axis in the reconnection region increase the distortion (Figure 9, right side), while for vanish-ing twist the distortion approaches zero as well (see Figure 10, left side). It is therefore hard to detect reconnection with this method for cases where the B

z component dominates.

More-over, other non-ideal processes may exist which show a similar pattern. For instance, if we use the same ,

W

and starting conguration as before, but lift the minimum ofW0toW0>0

on the z-axis, we model a non-ideal process which leads to a pile-up of magnetic ux and diusion but not to reconnection. The distortion for a minimum ofW0 of 0.2 instead of zero, is

shown on the right side of Figure 10 and can be compared with the reconnection case with a strongB

z-component at the left

hand side. Time=1. -0.5 0 0.5 y -0.5 0 0.5 x Time=1. -0.5 0 0.5 y -0.5 0 0.5 x

Figure 10: Distortion of a reconnecting ux tube with a dom-inating B

z-component (left) and a case without reconnection

(right).

6 Parallel electric elds

Another important aspect of magnetic reconnection at

B

6= 0 is

the existence of an electric eld component parallel to the mag-netic eld. The parallel electric eld may accelerate particles inside a reconnection region to high energies and is suggested to be responsible for fast particles observed in solar ares as well as auroral phenomena [3]. The maximum energy to which a particle can be accelerated is given by the line integral ofE

k

along the magnetic eld lines R E

k

dl across the non-ideal

re-gion. In Figure 11 this energy is plotted over the cross-section of our ux tube. The absolute value of the integrated E

k

de-pends on the reconnection rate (see Equation (8)) and the size of the reconnection region. The most astonishing feature of Figure 11 is the layer-like appearance of the structure. The distribution of the electric eld, although rather broad in the reconnection zone yields a extremely thin structure integrated along the magnetic eld lines. This is an eect of the nite

-1 -0.5 x0 0.5 1 -1 -0.5 0 0.5 1 y -1 -0.5 0 0.5 1 x -1 -0.5 0 0.5 1 y 0 1 2 3 4

Figure 11: The electric eld integrated along the eld lines. length of the reconnection site. It produces a distortion of the parallel magnetic eld in the transition zone where the poten-tial  decreases to zero. This distortion, shown in Figure 8, produces a major part of the mapping eect which leads to the extremely thin structure of the integrated parallel electric eld. There is an parallel electric eld associated with this distortion of the magnetic eld as well, but its contribution to the integrated electric eld is small.

7 Summary

The structure of the magnetic eld for localized reconnection events in B 6= 0 regions shows several aspects which can not

be explained by a simple superposition of well known two-dimensional models with null-lines and a constant component

of the magnetic eld parallel to the null-line. The localization implies additional magnetic structures in the transition zones localized above and below the part of the magnetic eld line where the reconnection occurs. In contrast to reconnection at magnetic null-linesB6= 0-reconnection shows no unique

signa-ture of magnetic eld line topology which allows for a sharp dis-tinction of reconnection from other non-ideal processes. How-ever, for weak

B

k-components the mapping of eld lines across

the localized reconnection region shows typical S-shaped dis-tortions which may used as an indicator of reconnection for these cases. The electric eld integrated along the magnetic eld lines across the reconnection region shows this S-shaped signature even more pronounced.

We also showed that a denition of reconnection is possi-ble on the basis of a covariant formulation of electromagnetic ux conservation. This denition, although rather abstract, also holds for cases with strong

B

kand is the straight forward

generalisation of the notion of two-dimensional magnetic recon-nection where the magnetic eld can be interpreted as frozen in a non-continuous velocity eld.

Acknowledgments

This paper is dedicated to Karl Schindler in recognition of his excellent work as a scientist and teacher. The work was sup-ported by the Deutsche Forschungs Gemeinschaft, (Projekt: Schi-156) and the British German Academic Research Collab-oration. We also gratefully acknowledge the helpful comments by Thomas Neukirch.

References

[1] Hesse, M. and K. Schindler, J Geophys. Res.

93

, 5559 (1988).

[2] Schindler, K. and M. Hesse: J. Geophys. Res.

93

, 5547 (1988).

[3] Otto, A. and G. Birk, Geophys. Res. Lett.

20

, 2833 (1993)

[4] Rastatter, L. and G. Hornig, this issue.

[5] Hornig,G. and K. Schindler, Astro. Lett. and Comm.

34

, 231 (1996).

[6] Hornig, G., submitted to Phys. Plasmas (1996).

[7] Hornig, G. and K. Schindler, Phys. Plasmas

3

, 781 (1996).

[8] Hornig, G. and L. Rastatter, to be published in Adv. Space. Res. (1996).

[9] Priest, E.R. and T.G. Forbes, J. Geophys. Res.,

97

, 1521 (1992).

[10] Priest, E.R. and P. Demoulin, J.Geophys. Res.,

100

, 23443 (1995).