5.1 Mapping discontinuities: Coronal fans and bald patches

It is possible for a continuous coronal field, one for which spatial derivatives of $B(x)$ are defined and bounded everywhere, to generate discontinuities in its footpoint mapping. One significance of such mapping discontinuities is that they are natural locations for the formation of magnetic discontinuities in an initially continuous field (Low, 1987 ; Low and Wolfson, 1988 ). While debate persists on the viability of magnetic discontinuities constrained by continuous mappings (see Section 3.3), there is no question that a discontinuous mapping is consistent with a discontinuous field. This means that if a field is initially continuous, but has a discontinuous footpoint mapping, then subsequent dynamics, even subject to strict line tying, may produce a discontinuous field. Indeed, analysis shows that line tied evolution will almost always lead to such discontinuities (Inverarity and Titov, 1997 ).

A mapping discontinuity is a curve across which neighboring footpoints, i.e. footpoints separated by an infinitesimally small distance, map to points a finite distance apart. This property means that spatial derivatives of the mapping functions $X+(x -,y -)$ and $Y+(x -,y- )$ are not defined at the discontinuity. Two types of structures in a continuous coronal field can lead to a discontinuous mapping. These are coronal null points and bald patches (Seehafer, 1986 ; Titov et al., 1993 ; Bungey et al., 1996 ), which are portions of PILs. In each case there is a surface in the corona, a separatrix, whose footpoints sit on the curve of mapping discontinuity (Bungey et al., 1996 ).

Two field lines in contact across the separatrix surface will, by definition, be rooted in different footpoints. They almost certainly also differ in other physical characteristics such as their total length or their end-to-end Alfvén transit times (Lau, 1993 ). Since these properties are important factors in the line-tied dynamics and equilibrium of a field, it is hardly surprising that magnetic discontinuities tend to form along these separatrices.

In most pointwise mapping models the vertical photospheric field $Bz(x, y,0)$ vanishes only along curves, i.e. PILs. Barring specially constructed cases there are no photospheric nulls in such models, since both components of the horizontal field will not, in general, vanish at the same point along a given PIL. Therefore, all null points in a pointwise mapping model will be located in the corona.

The fan of a coronal null will map to a photospheric curve which then defines a mapping discontinuity (see Figure 13 ). Footpoints on opposite sides of this curve will map to the ends of the null point’s two spines. One side of the discontinuity to the end of one spine, the other side to the end of the other spine. Points on the curve do not map to the photosphere at all, but end at the coronal null. The photospheric points to which the spines map will be singularities of the mapping, but not in the sense of a simple discontinuity: The neighborhood of this point will map to the entire region near the footpoints of the fan surface.

Figure 13:

Two-dimensional illustrations of field line mappings exhibiting both types of mapping discontinuities. In each case neighboring footpoints $a$ and $b$ map to points $a'$ and $b'$ separated by considerable distance. Each time the photospheric field is quadrupolar with 3 PILs indicated by vertical lines. The negative (downward) photospheric regions are shaded, and vertical and horizontal arrows show the sense of the photospheric field. Panel a: a field with a coronal null point. Although it is a two-dimensional illustration we take the null to be negative, with spines indicated by dark solid lines and fan field lines by thinner solid lines. Footpoints $a$ and $b$ map from opposite sides of the fan surface to points near each of the spine field footpoints. Panel b: a bald patch where a coronal field line (solid) grazes the photospheric surface, crossing in the inverse sense, from negative to positive, as indicated by the horizontal arrow.

Figure 14:

A schematic illustrating the photospheric features associated with a simple bald patch. The larger (lower right) figure shows the top-view of a sinuous PIL (solid) separating positive (white, $Bz > 0$ ) from negative (grey, $Bz < 0$ ) photospheric field. The bald patch is the darker portion extending between the two $×$ s: the points where $(B . \~/ )Bz = 0$ . The dashed curves ( $S+$ and $S -$ ) are footprints of the two separatrix surfaces. Three insets show elevation views of the BP separatrices (solid) and two field lines (dashed) which interconnect footpoints labeled as white or black circles. These views cut along the dotted curves in the main, plan view, along which the same circles show the footpoint locations. The lowest of these curves cuts near the center of the BP showing the correspondence with the two-dimensional version from Figure 13

The other type of discontinuity comes from a portion of the PIL called a bald patch. When the horizontal field at a PIL crosses from the positive ( $Bz > 0$ ) to negative ( $Bz < 0$ ), called the normal sense, the field lines will be concave downward. This produces a mapping which simply “flips” a neighborhood across the PIL - a continuous operation. The opposite situation, when the horizontal field crosses from negative to positive, called the inverse sense, produces concave upward field lines. The portion of the PIL where the horizontal field is inverse, and hence $(B .\ ~/ )Bz |z=0 > 0$ , is called the bald patch (BP). Unless the PIL is closed, and entirely inverse, the bald patch will extend between points where the horizontal field is parallel to the PIL, $(B .\ ~/ )Bz |z=0 = 0$ .

The mapping discontinuity occurs across the set of field lines which graze the surface at the BP (Seehafer, 1986 ; Titov et al., 1993 ). These field lines form two surfaces, the BP separatrices, departing the PIL horizontally in opposite directions (see Figure 13 ) and mapping to curves of footpoints in the photosphere. A simple bald patch, which may occur in a potential or a non-potential field, has the characteristic structure shown schematically in Figure 14 . The BP itself extends along the portion of the PIL between points where $(B . \~/ )Bz = 0$ , marked by $×$ s in the figure. The two BP separatrices extend upward from this curve forming open shells over the “normal” PIL (thin solid). These surfaces intersect the photosphere along two footprints denoted by dashed curves connected to the ends of the BP (their ends are marked with a $×$ and a $+$ ). Separatrix $S +$ has one footprint in the positive (white) region and maps to the negative (grey) side of the BP. Each BP separatrix has an open edge, the lip of the shell, extending between a $×$ and a $+$ as shown in the upper inset. The footpoint mapping will be discontinuous across the entire three-part curve: $S+$ -BP- $S-$ .

The BP separatrix is not a separatrix in the same sense as other separatrices, since it does not partition the field into separate domains or flux systems. It is evident from Figure 14 that the footprint of the separatrix $S+$ -BP- $S -$ is not a closed curve. This is a general property of bald patch separatrices, which therefore do not completely enclose coronal sub-volumes. While the field lines on opposite sides of a BP separatrix have distinct properties, there is often a continuous set of field lines between these two, passing around the separatrix. It is analogous to a fence which, however solid it may be, cannot effectively pen an animal since it is not completely closed.

BP separatrices from different BPs may intersect to form a separator (Bungey et al., 1996 ). The most common cases of BP separators occur where two BPs occur on the same PIL, such as in the field of Titov and Démoulin (1999 ). Unlike a null-null line, this separator is not associated with any null points, not even with submerged null points.

It is worth noting that the sense of concavity in a general field line is a geometric property, not a topological one. Locations of downward concavity, called “dips”, in field lines throughout the corona play a significant role in dynamic models of prominence formation (Tandberg-Hanssen, 1995 ). But a field line whose dip does not touch the photosphere can be continuously deformed into an undipped field line; the dip is therefore not a topological property. This is not possible when the dip grazes the photosphere since the photosphere is considered immovable. Thus a bald patch separatrix owes its infinitesimal thinness to the assumption that the photosphere is an infinitesimally thin surface. In models which treat the photospheric and chromospheric layers more realistically, BP separatrices become quasi-separatrices, defined similarly to quasi-separatrix layers in Section 5.2 (Karpen et al., 1990 ; Billinghurst et al., 1993 ; Lau, 1993 ).

Perhaps the most well-studied three-dimensional field with bald patches is an analytic model, proposed by Titov and Démoulin (1999 ), of a twisted flux rope nested under a potential arcade. The Titov and Démoulin field is a general force-free equilibrium produced by a superposition of a toroidal current ring (major radius $R$ and minor radius $a « R$ , with current uniformly distributed inside it), a submerged line current (running along the axis of the ring at depth $d$ ), and two submerged point sources (sitting on the line current and separated by $2L$ ; see Figure 15 ). It is a four-parameter class of equilibria after imposing force balance and leaving free the size and strength scalings. Titov and Démoulin (1999 ) studied a one parameter quasi-static emergence scenario where the major radius $R$ increases while its center’s depth $d$ remains fixed.

Figure 15:

The elements composing the Titov and Démoulin (1999

) model of a twisted flux rope under an overlying arcade. The figure depicts a current ring of radius $R$ , a line current at depth $d$ and a pair of point sources separated by $2L$ (reproduced from Titov and Démoulin, 1999

Figure 16:

Bald patches and their separatrices in the Titov and Démoulin (1999

) equilibrium. Bottom panels show the photospheric normal field (grey), PIL (thin black curve), BPs (thick black curves) and traces of the BP separatrices (white solid curves). Left is the state with one BP, $Ra < R < Rb$ , which naturally has two separatrix traces. Right is the state after bifurcation, $R > Rb$ , with two BPs and four separatrix traces. The top panels show perspective views of the BP separatrices from the two BPs of the bottom right case (reproduced from Titov and Démoulin, 1999

The photospheric field from the above construction consists of a positive and a negative region separated by a single sinuous PIL (see Figure 16

). Each region is concentrated where the current ring crosses the photosphere, giving the appearance of a classic bipolar active region. The field contains no coronal nulls, so any separatrices must originate in BPs. Titov and Démoulin (1999

) find that a BP forms at the center of the PIL when $R$ first exceeds a critical value $R = Ra$ . The BP then grows as its endpoints (where $B . \~/ Bz = 0$ ) move outward. The two separatrices from this single BP extend into the positive and negative region, respectively, where their footpoints (the separatrix traces) form an S-shaped (or inverse S-shaped) footprint when combined with the BP, as illustrated schematically in Figure 14

. At a second critical value, $R = Rb$ , the center of the PIL reverts to a normal sense, $B . \~/ B < 0 z$ , meaning that the BP has bifurcated into two portions. These two BPs have a total of four separatrices, two of which intersect to form a separator.

The Titov and Demoulin field represents a twisted flux tube underneath an arcade. While these are distinct elements in the field’s construction (a toroidal flux ring and a line-current, respectively) the resulting field cannot be unambiguously partitioned this way. Due to the open nature of a BP separatrix, discussed above, it does not separate the field into distinct flux systems which might be called “flux tube” or “arcade”. It is fair to say the field just beneath the BP separatrix is part of the flux tube, and that field just above is part of the arcade. This distinction becomes less clear, however, with increasing distance from the BP separatrix.

Titov and Démoulin (1999 ) propose that the actual coronal field due an emerging twisted flux rope would have a similar topology and geometry, including the S-shaped, or inverse-S-shaped BP separatrices. If and when the field became dynamically unstable, they went on to argue, strong currents would naturally form along the separatrix surfaces. Numerical simulations (Fan and Gibson, 2003 ) have confirmed that free dynamical evolution leads to current sheets along the BP surfaces in fields of the Titov and Demoulin type.

This theorized configuration could explain the occurrence of soft X-ray sigmoids prior to the onset of eruptive flares (Canfield et al., 1999 ). Observations show a strong preference for sigmoids to be S-shaped in the South and and inverse-S-shaped in the North. The Titov and Demoulin model attributes those shapes with flux ropes twisted in right-handed and left-handed helices, respectively, which are known to be the dominant magnetic chiralities in the South and North hemisphere (see, for example, Zirker et al., 1997 ).