List Of Figures

	Figure 1: Two-dimensional magnetic fields which are topologically equivalent. Curves show magnetic field lines, and the dark field lines are the separatrices from a magnetic null point (X-point). Panel a: a null point in a potential magnetic field. Panel b: a non-potential field with small scale structure and current, which is nevertheless topologically equivalent to the potential field in Panel a.
	Figure 2: An EUV image of the coronal plasma made by TRACE at $171 ºA$ on October 26, 1999. The intensity of $171 ºA$ emission from a small portion of the solar disk is indicated by a reverse color table (darker indicates higher emission). An inset shows the location of the irregular field of view on the solar disk.
	Figure 3: An illustration of how the ideal induction equation implies that the operations of tracing field lines and following trajectories commute with one another. Field lines (green lines) are tangent to the magnetic field vectors (green arrows), while trajectories (red lines) are tangent to velocity vectors (red arrows). These are followed beginning with the point $r0$ in both orders to the points $rmt$ and $rtm$ . The net displacements $d2rmt$ and $d2rtm$ are shown by blue arrows (they are not infinitesimal in the figure). In the case that $B$ and $v$ are related through the ideal induction equation, shown on the right, $rmt = rtm$ , so the order of operations does not matter.
	Figure 4: Schematic depictions of positive null points. Two spine field lines directed toward the null (white circle) appear as dark lines with arrow heads next to the null. The central portion of the horizontal fan surface is colored grey, and contains fan field lines directed outward like spokes on a wheel. The left case is the simplest: a potential null point which is cylindrically symmetric ( $c2 = c3$ ). Thinner lines show a few of the field lines on either side of the fan surface. The right case is a non-potential null whose fan is spanned by eigenvectors of complex eigenvalues and whose spines are not orthogonal to the fan surface.
	Figure 5: The structure of a null-null line. Positive and negative null points $B$ and $A$ have fan surfaces $SB$ and $SA$ shown in light and dark shades of grey, respectively. These intersect transversally along the null-null line shown as a thick black line. Surface $S B$ is then bounded by the two spines from null point $A$ (blue), and similarly for $SA$ and the spines from $B$ (red). The neighborhood of the null-null line is illustrated by the inset. This shows the direction of $B$ within a plane pierced normally by the null-null line at the dark circle. The fan surfaces cross the plane along the dotted lines.
	Figure 6: A tangential magnetic discontinuity of the Greene-Syrovatskii type, in a two-dimensional magnetic field. Lines show magnetic field lines which are contours of a flux function $A(x, y)$ , arrows indicate the field’s direction. The field direction reverses across the current sheet of width $D$ . The vertical field along the $x$ -axis is plotted at the bottom. The Y-type null points are located at the tips of the current sheet: $(x,y) = (0,± D/2)$ .
	Figure 7: Two small sections of a full-disk magnetogram made on 2000 March 17 by the SOI/MDI instrument on the SOHO spacecraft (Scherrer et al., 1995). The grey-scale shows the component of the magnetic field along the line of sight: Black is negative (away from the detector), white is positive, and grey is zero. Axes are labeled in arc seconds from the center of the solar disk. The top panel is a small ( $214 Mm$ on a side) region of the quiet Sun. Black and white specks are unipolar magnetic elements, each roughly $3 × 1018 Mx$ , with maximum field strengths $\|Blos\|~ 150 G$ ; the grey-scale extends from $-150 G$ to $+150 G$ . Bottom is a small, young active region (NOAA 8910) plotted on a grey-scale extending from $- 1000 G$ to $+1000 G$ .
	Figure 8: A version of Sweet’s original model of four interacting flux domains (cells) from four discrete photospheric sources. The top panel shows the locations of the 2 positive (white) and two negative (black) sources. The two magnetic null points, $B1$ and $A2$ are shown by triangles. Dashed and solid lines are fans and spines, respectively. On the bottom is a perspective view of one representative field line from each of the four flux domains: $P 1$ - $N 3$ and $P 2$ - $N 4$ (red) and $P 1$ - $N 4$ and $P 2$ - $N 3$ (green). The blue line is the field’s separator, running from the positive to negative null. Black lines are the spines from the two nulls.
	Figure 9: Left: The footprint of a potential field from 6 photospheric sources, labeled, e.g., $P 1$ , $P 2$ , etc. The 4 null points are labelled $B1$ , $B2$ , etc. Fan traces and spines (dashed and solid curves, respectively) divide the plane into 7 domains, which are labeled $P 1$ - $N 4$ , $P1$ - $N 6$ , etc. On the right is a schematic depiction of the connectivity, called a domain graph.
	Figure 10: The two possible connectivities of Sweet’s configuration. Footprints are shown on the left and corresponding domain graphs on the right. The top and bottom rows are cases A and B, respectively, of Gorbachev et al. (1988). Case A (topologically equivalent to Figure 8) has four domains, and the fan surfaces from the two nulls interect along a separator (not shown), so both fans are broken fans. For example, the fan traces from $B1$ ( $\~/$ ) connect to $N 3$ (downward) and to $N 4$ (upward, although the complete fan trace is not shown). In case B (bottom row) the fans from null points $B1$ and $A2$ are unbroken enclosing domains $P 1$ - $N 3$ and $P2$ - $N 4$ , respectively. For example, both fan traces (dashed lines) from null $B1$ connect to $N 4$ . A potential field will switch from case B to case A through a global separator bifurcation as sources $P 2$ and $N 3$ approach one another.
	Figure 11: An example of a global spine-fan bifurcation. The two panels show a portion of the footprint of the field before (left) and after (right) the bifurcation. The bifurcation occurs as the spine (solid curve) of null point $B1$ sweeps across the fan (dashed curve) of null point $B2$ . As a consequence the spine source of $B1$ switches from $P 1$ to $P 6$ , and the fan trace of $B2$ sweeps from $N 5$ to $N 4$ . At the instant of bifurcation (not shown) the spine from $B1$ is part of the fan of $B2$ .
	Figure 12: The footprint of a potential field modeling the lower left quadrant of the quiet Sun field shown in Figure 7.
	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.
	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).
	Figure 17: QSLs in a submerged pole model of Sweet’s configuration. Contours shows the photospheric field $Bz(x, y)$ , with the dark line designating the PIL (here labeled IL). (Left) Grey scale shows $Q(x)$ on a logarithmic scale. (Right) Grey scale shows the degree of flux tube expansion for each footpoint. Plusses and dots indicate the principal locations of the four interacting flux systems. (Reproduced from Titov et al. (2002).)
	Figure 18: Two submerged poles models of AR 2776, November 5, 1980. The grey scale shows the vertical magnetic field at the photosphere from the collection of submerged poles. The top panel shows the 4-charge co-planar model by Gorbachev and Somov (1989). The bottom panel shows the 18-pole model by Démoulin et al. (1994). The axes are labeled in units used in that paper. The projected location of each charge is indicated. The PIL is a thin broken line, and several separatrix traces are shown as solid and dashed dark curves. The solid curve is the first intersection of the fan surface from the submerged null (triangle), the dashed curve is its second crossing. Thin solid lines show the sub-photospheric spines of the selected null points.
	Figure 19: One of the separatrices in the 18-source model of Démoulin et al. (1994). A portion of the fan surface of a submerged negative null point (triangle), and its spine curves (solid), one extending to $N 17$ and the other leaving the box. The box is a section of the region below the photosphere, showing charges at their respective depths as $+$ s and $×$ s. Dashed lines show field lines from the null’s fan surface extending upward to the photosphere. A dark solid curve indicates where they cross $z = 0$ , and the thin solid lines are the same field lines above the surface. A dark dashed line shows where the separatrix descends again below the photosphere, mapping to sources $P 8$ , $P15$ and $P 16$ . These same photospheric curves appear in the lower panel of Figure 18.
	Figure 20: The footprint of the 4-charge co-planar model of Gorbachev and Somov (1989) after source-depths are taken to zero. Symbols are the same as MCT footprints. Dotted lines are the field’s two separators. The top panel of Figure 18 is the central portion of this footprint. This is identical to Figure 4 in Gorbachev and Somov (1989).
	Figure 21: A local separator bifurcation. Footprints of the potential field generated by two configurations. The left configuration has two positive prone null points, B1 and B2, whose fan traces asymptote to the same line. In the right configuration one positive source, P1, has been moved to the right (arrow) leading to a local separator bifurcation. This bifurcation creates a new positive prone null, B4, and a negative upright null, A3, whose coronal spine is shown as a dotted line. Fan traces from B1, B2 and B4 all connect to the upright null A3.
	Figure 22: A local double separator bifurcation. Panel a: a single prone null located at the origin. The spines (solid dark line) run along the $x$ axis, lying in the photospheric plane (horizontal dotted square). Thin lines show a selection of fan field lines in the corona ( $z > 0$ ), mirror corona ( $z < 0$ ) and in the photosphere (i.e. the fan traces). Panel b: the structure of the field after bifurcation. The null at the origin has reversed sign, so its spines now form the $x$ axis. New nulls now appear above and below the origin on the $z$ axis. Their fans lie in the $y = 0$ plane, and their spines extend horizontally in the $± ^y$ direction. These spines form the top and bottom edges of the fan surface of the prone null. The three fans intersect along a pair of separators running along the $z$ axis between the null points (reproduced from Brown and Priest, 2001).
	Figure 23: Possibilities for topological boundaries with open field lines. Lower panels show the photospheric field in relation to the PIL (dark solid curve) and separatrices (dashed). In each case, the positive field ( $+$ s) forms a compact region separated from the surrounding negative region by the PIL. Top panels are elevation views of the field lines (thin curves) and separatrices (dark curves) anchored to the positive and negative regions denoted with $+$ s and $-$ s. Panel a: a closed separatrix enclosing a compact region of closed field. Panel b: a “crater-like” enclosure where the closed field region surrounds open field (reproduced from Yeh, 1978).
	Figure 24: A schematic depiction of a source-surface model as viewed from above the Sun’s North pole (the sense of rotation is indicated by a semi-circular arrow). A dashed circle shows the source surface at $r = RS$ . The field is made purely radial at this surface by setting $Bh = Bf = 0$ . Between the source surface and the solar surface, $r = Ro .$ , the magnetic field is potential: $B = - \~/ x$ . Field lines anchor to the photosphere in a negative region (shaded segment) and positive regions. Outside the source surface the field is swept back in a Parker spiral. Two null points (X-points) are shown as circles on the source surface. The upward and downward separatrices are shown in blue and red, respectively. The sector boundaries are shown as green curves.
	Figure 25: A schematic depiction of the sector boundary defined by the potential source surface model. On the left is a view of the coronal field, $R o. < r < RS$ , in the meridional plane using the same scheme as Figure 24. The X-points at the source surface are shown by circles. From them originate one downward separatrix (red) and one upward separatrix (blue). The map on the right is the source surface itself ( $r = RS$ ) plotted as sine of latitude versus longitude. The dark curve is the sector boundary, $BS = 0$ , separating the outward sector ( $BS > 0$ , white) from the inward sector ( $BS < 0$ , grey). The dashed vertical lines at $0o$ and $180o$ correspond to the two meridional slices on the left.
	Figure 26: Examples of particularly complex sector boundaries. These are contour plots of $B (h,f) S$ at the source surface, $R = 2.5R S o .$ , versus Carrington longitude (horizontally) and latitude (vertically) similar to the format of Figure 25. Dark solid curves show the sector boundary, $BS = 0$ , separating inward sectors (partially grey with red contours) from outward sectors (blue contours). The top panel is from September 1999, as the Sun’s dipole is reversing and the quadrupole moment is dominant. There are four sector boundary crossings along the solar equator. The bottom panel is from January 2002, when the Northern hemisphere is mostly inward but includes a patch of outward flux enclosed by a second sector boundary. (Courtesy of J.T. Hoeksema and Wilcox Solar Observatory).
	Figure 27: A schematic depiction of the coronal hole boundaries defined by the source surface model. The left is the same meridional view as in Figure 25. The right part now shows the solar surface ( $r = Ro .$ ) plotted as sine of latitude versus longitude. The blue and red curves are the footprints of the upward and downward separatrices, respectively; the black curve is the PIL. The shaded regions are (from top to bottom) the outward coronal hole (light grey), positive closed flux (white), negative closed flux (grey) and inward coronal hole (dark grey). The same colors on the left part indicate how these regions might appear on the disk.