4.1 The photospheric field

The extended photospheric region with no normal field, $Bz = 0$ , heretofore called “field-free”, plays an important role in MCT models. Taking the limit of the coronal field just above gives values for $Bx(x, y,0)$ and $By(x,y, 0)$ within the plane. Alternatively, one may define the photospheric field by reflecting the coronal field into the half-space $z < 0$ (the mirror corona) and requiring continuity everywhere outside photospheric source-regions. Either construction shows that, in general, $Bx(x,y, 0)$ and $By(x, y,0)$ vanish only at isolated points called photospheric null points.

Since $z = 0$ is a plane of symmetry, field lines will remain within it, and they will have a two-dimensional topology of their own. The topology of photospheric fields was first analyzed in a methodical fashion by Molodenskii and Syrovatskii (1977 ). Recalling that the photospheric plane actually represents the merging layer, we can equate its topology with the topology of chromospheric features such as $Ha$ fibrils (Filippov, 1995 ). The topology of the photospheric field is characterized by the footprint (Welsch and Longcope, 1999 ), showing the sources ( $+$ s and $×$ s for positive and negative), the photospheric null points ( $\~/$ s and $/_\$ s for positive and negative nulls) along with their spines (solid lines) and the photospheric lines of their fans (dashed lines). Figure 8 shows the footprint of Sweet’s configuration in the grey field-free sea, while Figure 9 shows the footprint of a slightly more complex example.

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.

Due to reflectional symmetry $^z$ must be an eigenvector of a photospheric null point’s Jacobian matrix. This vertical eigenvector will be either the spine or part of the fan, making an upright or a prone null, respectively (Longcope and Klapper, 2002

; Beveridge et al., 2003

). Prone nulls form hyperbolic (saddle) points in the photospheric field; they resemble two-dimensional X-points but are in fact generic three-dimensional nulls. The four photospheric field lines connecting to a prone null include two spines and two field lines from the fan, called fan traces. These are rendered in a footprint diagram, such as Figure 9

, by solid and dashed lines respectively. Due to their vertical orientations upright nulls have no spines or fan traces in a footpoint diagram. The may, however, connect to spines or fan traces from prone null points.

The photospheric magnetic field is a two-dimensional vector field with sources, sinks and hyperbolic saddle points. Positive sources and positive upright nulls are sources, negative sources and negative upright nulls are sinks, and prone nulls are saddle points. The number of sources $S$ (including $oo$ if the net charge is not zero) is related to the number of upright and prone nulls $nu$ and $np$ through the Poincare index theorem (Molodenskii and Syrovatskii, 1977 ; Inverarity and Priest, 1999 ),

This relationship is invaluable for analyzing the topology of the photospheric magnetic field.

The footprint is divided into domains by the spines and fan traces of prone nulls. Any prone null whose fan traces both go to the same source is an unbroken fan which will most often enclose a single domain. For example, null $A4$ in Figure 9 has an unbroken fan enclosing domain $P 1$ - $N 5$ . Each spine and fan trace from the remaining prone nulls are edges of this partitioning. Euler characteristic of this construction shows the number of photospheric domains to be

(Longcope and Klapper, 2002

), when there are $nuf$ nulls with unbroken fans, assumed to each enclose a single domain, or to recursively enclose other domains of unbroken fans. For example, Figure 9

has $n = 4 p$ nulls, of which only $A4$ has an unbroken fan, $n = 1 uf$ , therefore there are $D = 7 f$ domains in the footprint.

When all nulls have unbroken fans, Equation (16 ) yields $Df = np$ , which is too small by one. Eliminating the unbroken fans in this pathological case leaves a trivial configration with two sources, one domain and no edges, not consistent with Euler’s characteristic.

Upright nulls of a given sign seem to occur most frequently surrounded by sources of the opposing sign. A study of potential fields generated by uniform, random distributions of point sources shows that the density of upright nulls is proportional to the density of sources. The constant of proportionality depends on the distribution of source magnitudes and on the imbalance of flux in each sign, reaching a maximum of $0.03$ when all sources are of the same sign and magnitude (Beveridge et al., 2002 ). Using this in Equation (16 ) shows that the density of prone nulls will be $-~ 1.1$ times the density of sources in the case with only one sign of source.