CURRENT SHEET FORMATION AND RECONNECTION ON
SEPARATOR FIELD LINES
D.W. LONGCOPE
Montana State University
Department of Physics
Bozeman, MT 59717-3840
Abstract. Magnetic reconnection is believed to play a fundamental role
in many types of coronal activity such as aring and transient loop brightenings. Reconnection occurs when ux is transferred between neighboring
coronal ux systems. A single ux system consists of coronal magnetic eld
lines originating in a common pair of photospheric features, (i.e. ux tubes).
Changing the ux in such a system requires a substantial electric eld in
the corona, parallel to a magnetic eld line in at least one place. Such an
electric eld is not consistent with the highly conductive nature of the hot
corona.
We present a model for the evolution of a complexly structured coronal magnetic eld driven by motion of photospheric ux tubes. Barring
reconnection, motion of photospheric ux tubes leads to the development
of three dimensional current sheets (current ribbons) along particular eld
lines called separators. A separator is a eld line which lies at the boundaries of four distinct ux systems simultaneously. The amount of current in
a given ribbon is an increasing function of ux tube displacement. The presence of current implies free magnetic eld energy, and reconnection must
occur eventually if the separator current and stored energy are not to grow
indenitely. When it occurs, reconnection adjusts the uxes in each system
to temporarily obviate the need for current ribbons, thereby liberating the
stored free energy. Such instances of reconnection occur sporadically during
a typical evolutionary scenario. The frequency of these events, and the energy released, is consistent with transient brightenings observed by Yohkoh
(Shimizu et al., 1992).
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D.W. LONGCOPE
Figure 1. Synthetic magnetograms of the two bipoles shown with x running vertically
and y horizontally. Here y = d. Three cases are shown: (a) x = 0, (b) x = 0:75 d
(c) x = 1:5 d.
1. Quantifying ux interlinkage
To introduce our model we will consider a case of two interacting magnetic
bipoles. This case, considered in many previous investigations (Sweet, 1958;
Baum and Bratenahl, 1980; Gorbachev and Somov, 1988; Longcope, 1996),
is one of the simplest with enough geometric complexity to manifest generic
features of three dimensional magnetic elds such as a separator. The conguration consists of four magnetic poles (\charges") P 1, P 2, N 1 and N 2
each with ux 0 , arranged in two bipoles (see Fig. 1). The poles interact
with one another as the distance between bipoles, x, changes due to some
relative photospheric motion.
The magnetic poles represent the tops of magnetic ux tubes. These are
conned to a tube below the canopy by ows and plasma pressure, but in
the corona the eld expands into a space-lling equilibrium (see e.g.Gabriel
1976 ). Almost all of the coronal eld lines will connect a positive (P ) charge
to a negative (N ) charge. The set of eld lines connecting, for instance, pole
P 2 to N 1 will have some total net ux 21. This ux can be calculated by
integrating the vector potential around a path C which encloses only those
eld lines
I
21 = A dl :
(1)
C
CURRENT SHEETS AND RECONNECTION ON SEPARATORS
3
Figure 2. The separator connecting null points B and A (solid), along with a possible
closure (dashed) formed by projecting onto z = 0. Also shown are some of the P 2{N 1
eld lines.
This path must lie at the boundary separating those eld lines which connect P 2 to N 1 from all other eld lines. There is a single eld line, called
the separator , with this property (Longcope, 1996). We will take C to
consist of the separator eld line and some curve lying just below the z = 0
surface (see Fig. 2).
The value of 21 will depend on the conguration of poles, and on the
form of the coronal magnetic eld. Taking the coronal eld to be currentfree (its state of lowest energy) yields a curve (v)
21 (x) shown in Fig. 3. The
ux is zero for x = 0, due to symmetry about the x axis, and increases
up to x ' 1:33 d where P 2 and N 1 are very close to one another. For still
larger x the bipoles become increasingly isolated and (v)
21 ! 0.
2. Response to ux motion | Current ribbons
Changing the ux 21 would require an electric eld along C . Since the
bottom of this path lies within a conductor (z = 0) this electric eld must
exist along the separator eld line in the corona. Using typical values for
0 , d and d(x)=dt gives an average eld of 0:1 V =m, which is much larger
than the highly conductive corona could support parallel to the magnetic
eld. We must conclude that the corona cannot remain current-free in the
face of ux tube motion. Instead the coronal eld is a sum of the vacuum
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D.W. LONGCOPE
Figure 3. (b0 The separator length L scaled to the bipole separation d, and (B) the
ux (v)
0 , the ux of a single pole. Its maximum value is 0:34; one-third of
21 scaled to
the ux from P 2 links to N 1, the remining two-thirds links to N 2.
eld from the point charges and a contribution, vanishing at z = 0, from
coronal currents. The change in vacuum ux (v)
21 must be canceled by
self-ux from the coronal currents. A careful analysis of this problem by
Longcope and Cowley (1996) lead to the conclusion that the coronal current
forms a singular ribbon paralleling the separator. The current will ow
from one separator footpoint to the other, through the corona, and return
in a distributed current layer just beneath the canopy (z = 0). Since the
separator lies at the boundary of ux systems, current can close through the
canopy without crossing any eld lines, in contrast to models with current
owing on ux tubes themselves(Melrose, 1995).
The formation of a current ribbon on a separator is directly analogous
to the formation of a current sheet at an X-point in a two-dimensional
magnetic eld (Syrovatskii, 1971). In the case of a generic three-dimensional
eld the separator eld line plays the same role as an X-point in twodimensions (Greene, 1988; Lau and Finn, 1990).
The net coronal current is determined by the requirement (cr)(I ) =
?(v)
21 . The relationship between current and self-ux is essentially one
of self-inductance L. The (dierential) self-inductance will have the form
(v)
(2)
L = 1 d 2L ln(L=w) ;
c dI
c2
where w is the breadth of the current sheet which is a property of the
detailed structure of the singular magnetic equilibrium. p
For small I the
current will distort the equilibrium only locally and w jI j by analogy
to two dimensions (Longcope and Cowley, 1996). The constant of proportionality depends on the local gradient of the vacuum magnetic eld at the
CURRENT SHEETS AND RECONNECTION ON SEPARATORS
5
separator. This gradient can be used to dene a function I ?(x) according
to which the approximate current-ux relationship is
h
?
3
i
I e
ln
(cr) (I ) = LI
(3)
c
jI j 256 :
The exact denition of I ? , depending only on the current-free eld for x,
is given in Longcope, 1996.
3. Eects of reconnection | microaring
Changing x will give rise to a coronal current I owing roughly parallel
to the separator. This will store free energy (i.e. in excess of the vacuum
eld energy) as for any inductor. This energy is not localized to the sheet;
it corresponds to the extended magnetic eld due to the coronal current.
We propose that some upper limit exists to the current in the ribbon.
This limit might be due to instability at large or small scales, but can
be phrased as a threshold: jI j I ? , where is some number probably
less than unity. When this threshold is exceeded we assume that a large
electric eld occurs along the current ribbon (stick-slip reconnection). This
changes the ux 21 , bringing it closer to the present (v)
21 (x) and thereby
reducing I . Thus, some of the ux linking P 2{N 2 is transported across
the separator and remapped so as to link P 2{N 1: this is reconnection.
Complete reconnection takes 21 ! (v)
21 , eliminating I entirely, thereby
releasing all the free magnetic energy it stores. This, we propose, is the
underlying cause of sudden coronal energy releases, such as transient X-ray
brightenings (Shimizu et al., 1992).
Applying this model to the above conguration gives the time history
shown in Fig. 4. The threshold parameter has been set at = 0:15, and
physical values are those of relatively small bipoles, 0 = 3 1020 Mx,
d = 30; 000 km, which are moving moderately rapidly d(x)=dt = 300 m=s.
In the resulting evolution there are a series of small (E 1027 ergs)
events occurring at intervals of 3{6 hours while the bipoles interact. After
this initial phase there is a lull, followed by a longer period of smaller, less
frequent events as the bipoles become increasingly isolated.
4. Discussion
The example above shows how a coronal eld originating in multiple distinct
magnetic sources will evolve. We have chosen a simple case, with only two
sources of each polarity, which contains only a single separator. Motion of
the sources leads to coronal current which ows in a thin ribbon along the
separator. Cases with more magnetic sources have more separators, forming
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D.W. LONGCOPE
Figure 4. The time history of the evolving bipoles. (a) The ux 21 (solid) which attempts to follow (v)
21 (dotted) through a series of reconnection events. (b) The free-energy
from the current. This drops to zero with each reconnection event. (c) The amplitude of
the drops in energy. This is the energy liberated by reconnection.
a rich array of coronal current paths. Motion of these sources would result
in sporadic reconnection on many of the separators. Each reconnection
event would liberate energy causing a transient loop brightening there. The
resulting picture, an array of loops sporadically brightening, is reminiscent
of the now familiar Yohkoh SXT movies.
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