Upper and lower nulls

Figs. 2.1, 2.2 and 2.3 show the nature of the magnetic field with B0 = L0 = L= 1 and

various values ofa,bandcas detailed in the figure captions. Only features of the magnetic skeleton are plotted in order to highlight the topologically distinct flux domains.

In these figures, the location of the positive (lower) and negative (upper) nulls are identified by blue and red spheres respectively. The spines are denoted by thick lines with the same colour as their null spheres. The field lines in the separatrix-surface plane of the lower null are pale-blue and pink for the upper null.

The separatrix-surface plane equation for the lower null is simply y = 0 and the separatrix-surface plane equation for the upper null is

y= 2bL

aL−2−2cL+p(aL−2−2cL)2_{+ 4b}2_L2x. (2.11)

Fig. 2.1 shows how the geometry of the field lines of the lower and upper nulls are changed when ais increased. In this figureb= 2.0,c= 0.5 andais varied from 0.5 to 1.3 to 2.1.

For the lower null it is evident that the major axis of the separatrix surface is affected by a as can be seen from Eq. 2.5. When a= 0.5 the eigenvector associated with λf1 is

the major axis of the separatrix-surface plane sinceλf1 > λf2 (Fig. 2.1a). However, when

a >1, as seen in Fig. 2.1b where a= 1.3,λf2 > λf1 and the separatrix-surface field lines

start to align with the z-axis. This is more pronounced in Fig. 2.1c where a= 2.1. For the lower null, the value ofahas no effect on the plane of the separatrix surface, just the geometry of the field lines in this surface. Changingahas an effect on both the geometry of the field lines in the separatrix surface and the plane of the separatrix surface in the

xy-plane of the upper null.

Fig. 2.2 shows how the magnetic field configuration changes when b is varied. In this figure a = 1.0, c = 2.0 and b is varied from -6.5 to 1.5 to 8.5. From these images it is evident that b rotates the separatrix surface of the upper null in the xy-plane and also changes the geometry of the field lines in the separatrix-surface. The geometry of the lower null’s field lines is also changing as it is affected by the upper null although very close to the null these field lines are unaffected byb.

Finally Fig. 2.3 shows how the magnetic field behaves whenc is varied. In this figure

a= 1.0, b= 1.5 and c is varied from 0.1 to 1.1 to 2.1. As for the cases of varying aand

2.2. POTENTIAL SEPARATOR MAGNETIC STRUCTURES 33

nulls and also rotates the separatrix surface of the upper null in thexy-plane. Since these nulls are potential when the separatrix surface of the upper null rotates the spine must also rotate to stay perpendicular to the separatrix surface.

Figure 2.1: Images of the magnetic skeleton for our analytical magnetic field which includes two nulls joined by a separator. The nulls are identified by blue/red spheres for the positive/negative nulls, the separatrix-surface field lines by pale-blue/pink lines and the spines as blue/red thick lines for the positive/negative nulls. The separator is drawn here as a green line. The parameters of the magnetic field areb= 2.0,c= 0.5 and (a)a= 0.5, (b)a= 1.3 and (c) a= 2.1.

Figure 2.2: Images of the magnetic skeleton for our analytical magnetic field which includes two nulls joined by a separator. The nulls are identified by blue/red spheres for the positive/negative nulls, the separatrix-surface field lines by pale-blue/pink lines and the spines as blue/red thick lines for the positive/negative nulls. The separator is drawn here as a green line. The parameters of the magnetic field area= 1.0,c= 2.0 and (a)b=−6.5, (b)b= 1.5 and (c) b= 8.5.

2.2. POTENTIAL SEPARATOR MAGNETIC STRUCTURES 34

Figure 2.3: Images of the magnetic skeleton for our analytical magnetic field which includes two nulls joined by a separator. The nulls are identified by blue/red spheres for the positive/negative nulls, the separatrix-surface field lines by pale-blue/pink lines and the spines as blue/red thick lines for the positive/negative nulls. The separator is drawn here as a green line. The parameters of the magnetic field area= 1.0,b= 1.5 and (a)c= 0.1, (b)c= 1.1 and (c) c= 2.1.

Figure 2.4: Images of both nulls joined by a separator (green line) with field lines drawn in the separatrix-surface plane in pale-blue for the lower null and pink for the upper null. Extra field lines are also drawn here out along the spines in green for the lower null and yellow for the upper. The spine of the lower null is blue and the upper null’s spine is red. (a) Angled view. (b) Image shown in the xy-plane. Here a= 0.5, b = 2.0,c = 0.5,

2.2. POTENTIAL SEPARATOR MAGNETIC STRUCTURES 35

Fig. 2.4 shows the field lines drawn in the separatrix-surface plane and also field lines drawn out along the spines of the nulls. The extra field lines drawn out along the spines are shown in mauve for the lower null and orange for the upper null. These figures highlight the behaviour of field lines in the vicinity of the magnetic skeleton: field lines close to the spines or separatrix surfaces of the nulls follow the same paths.