In the idealized models of coronal field topologies treated here, coronal and photospheric nulls form distinct classes. Photospheric nulls only occur, in general, in the field-free photospheric regions of MCT models, and are absent from models with non-intermittent photospheric field. Coronal null points, on the other hand, are possible in all coronal models.
Enumeration of coronal null points in a given field is considerably more difficult than the enumeration of photospheric nulls. Formulae relating to photospheric null points, discussed in Section 4.1, follow from an application of Poincaré’s index theory to planar vector fields (Molodenskii and Syrovatskii, 1977). The sum of the indices of the sources (), upright nulls () and prone nulls () over the entire plane must equal the index of the asymptotic field which is either 1 or 2 for a field which is asymptotically monopolar or dipolar, respectively. Adding to the count one source at infinity in cases of monopolar field leads to Equation (16) for the number of photospheric null points of each type.
The topological degree (Molodenskii and Syrovatskii, 1977; Greene, 1993), which is the three-dimensional analog of Poincaré index, is somewhat less useful for enumeration of null points. A positive source and a negative null point have the same topological degree, , while negative sources and positive nulls each have topological degree . A volume containing positive/negative sources and positive/negative nulls will therefore have a topological degree . The topological degree can be established independently using only the field at the volume’s outer surface (Greene, 1993; Inverarity and Priest, 1999). A general formula follows from using a surface at infinity and evaluating the degree of the asymptotic field; this is , or 0 when the net charge is positive, negative or zero. Unfortunately, this procedure requires enclosing both the corona and its mirror image, so the general formula (Inverarity and Priest, 1999)
In a field undergoing topological changes, either due to non-ideal evolution (see Section 2.5) or hypothetical parameter variation, null points are created and destroyed only through local bifurcations. According to the general theory (see, for example, Guckenheimer and Holmes, 1983) singular points of a divergence-free field may change either through saddle-node bifurcation or, in cases of symmetry, through pitchfork bifurcation or Hopf bifurcation.
In a saddle-node bifurcation two singular points of opposite degree are created simultaneously. In the reverse bifurcation the two annihilate one another. A saddle-node bifurcation in a magnetic field, called a local separator bifurcation (Brown and Priest, 2001), creates one positive and one negative null. The two spines and the line of initial separation are all three mutually orthogonal. This means that there is a separator connecting the two nulls immediately following their formation. A local separator bifurcation can occur in the corona, where it automatically satisfies Equation (26), or within a field-free portion of the photosphere, where it creates one prone null and one upright null of the opposite sign in order to satisfy Equation (16) as well (see Figure 21).
A pitchfork bifurcation occurs only within field-free photospheric regions, and therefore is only relevant to MCT models. This bifurcation involves the transformation of one null into three, two with the same sign as the original null. Such a scenario is structurally unstable under general conditions where it would prefer to be a saddle-node bifurcation in the vicinity of an existing null point. In cases where symmetry forbids this generic version, however, such as at the plane, the bifurcation must occur as a pitchfork. In MCT models, where it is called a local double separator bifurcation (Brown and Priest, 2001), a prone photospheric null point, say it is positive, will bifurcate into a negative prone null, and a positive coronal null. The third null is the mirror image of the coronal null which is therefore also positive, making the total, two positive and one negative, in compliance with Equation (26). The spines of the coronal null will be parallel to those of the original photospheric null, and orthogonal to those of the new photospheric null (see Figure 22). The fan of the new photospheric null will follow underneath the coronal null’s spines.
There is one final mechanism which, while not a proper bifurcation, transforms photospheric nulls into coronal nulls, while observing Equations (16, 26) and reflectional symmetry. A prone and an upright null of the same sign can collide and “scatter” from the photosphere into the corona and mirror corona (Graham Barnes, private communication). Both the coronal null and its mirror image have the same sign as the original pair, so Equation (26) is obeyed. The simultaneous loss of a prone and an upright null from the photosphere satisfy Equation (16).
General MCT models have far fewer coronal nulls than photospheric nulls. Coronal nulls require complex source configurations. For instance they will not occur in distributions where it is possible to draw a straight line dividing the positive from the negative sources (Seehafer, 1986). This means that a minimum of four sources are required. Typical is a kind of -spot configuration with one source surrounded by three of opposite sign. This leads to a coronal null of the same sign as the central source whose fan surface forms a dome.
Nulls in submerged poles models are either sub-photospheric or coronal; only accidentally will one occur exactly at . The sources in these models are not co-planar, but restricted to a sub-photospheric layer. The likelihood of coronal nulls therefore depends on the extent to which nulls extend outside the source-layer (Bungey et al., 1996). Démoulin et al. (1994) posed the question “Are magnetic nulls important in solar flares?”, and ultimately concluded the answer must be “No”. In this work they used various placements and strengths of four submerged poles to model a variety of flaring active regions. They found that null points occurred above the photosphere very rarely, and that their inferred presence or absence was unrelated to properties of the flares observed.
There is some observational evidence for coronal null points especially above a photospheric magnetic concentration surrounded by opposing polarity. Filippov (1999) reported EUV observations (171 Å and 284 Å from SOHO/EIT) of AR8113 close to the west limb in which loops exhibited a very clear “saddle” configuration. The shape of the coronal loops suggested a coronal null point located approximately 65 Mm above a small positive region which had emerged into a dispersed negative polarity. There is no evidence in the EUV data for energy release or reconnection at this null point.
There is also evidence that magnetic null points do play a role in a few solar flares. Fletcher et al. (2001) study an M-class flare which occurred in AR8524 on 3 May, 1999. Magnetograms shows that a small bipole emerged in the trailing (negative) polarity of the existing active region just prior to the flare. In a potential field extrapolations from a point source representation of the magnetogram there is a coronal null point just above the surrounded positive pole; this feature is, however, absent from extrapolations which use the full magnetogram. Based on this and the morphology of TRACE EUV (171 Å) and Yohkoh SXT observations, Fletcher et al. (2001) conclude that the flare was initiated by reconnection at the coronal null point.
Evidence is also found, by Aulanier et al. (2000), for a coronal null at the initiation of the flare on July 14, 1998 (the first Bastille Day flare). The magnetic configuration in which this occurs (AR8270) is a -type sunspot. These are often characterized by a surrounded polarity, and as a general configuration they are well known to produce the largest solar flares (Zirin and Liggett, 1987). In this case, Aulanier et al. (2000) perform potential extrapolations from line-of-sight magnetograms (KPNO) and find a null point in the corona.
In models of random magnetic fields with homogeneous isotropic statistics it is possible to calculate an average density of null points. The density of null points in a general field with three-dimensional homogeneity depends on the spectral energy density (Albright, 1999). Null points will have a volume density
A coronal magnetic field is unlikely to be entirely homogeneous since it is anchored to the photosphere. For a potential-field extrapolation from the scale of structuring gets progressively smoother with height, causing the null density to fall dramatically (Schrijver and Title, 2002). For a homogeneous photospheric field composed of an equal mixture of positive and negative elements the null density has the universal form , independent of the sizes or density of the photospheric elements (Longcope et al., 2003). When the mixture is uneven the nulls become restricted to a thin layer, but with slightly higher overall column density. In most cases there is roughly one coronal null point for every ten photospheric sources (Schrijver and Title, 2002; Longcope et al., 2003). This means that nulls are relatively rare in the corona and get rarer still as one goes higher.
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