Initial and equilibrium magnetic skeleton

The eigenvalues and eigenvectors for the lower and upper nulls, with b=0.75,c=0.25,

jsep= 1.5 and B0 =L0 =L= 1.0, are as follows

λsl=− a 2 − p (a+ 2)2_−2.25 2 , esl= 1.5 a+ 2 +p(a+ 2)2_−2.25,1,0 T , λf1l=a, ef1l = (0,0,1) T_, λf2l=− a 2 + p (a+ 2)2_−2.25 2 , ef2l = 1.5 a+ 2−p(a+ 2)2_−2.25,1,0 T , λsu= 1.25, esu= 3.75 a−2.5,1,0 T , λf1u =−a, ef1u= (0,0,1) T_, λf2u =a−1.25, ef2u = (0,1,0) T_{. (4.3)}

4.3. MHS EQUILIBRIA A: THE EFFECTS OF VARYING PARAMETERA 87 Increasing the parameterabrings the spine and the separatrix surface of the lower null closer to being aligned along they-axis and to the yz-plane, respectively. The spine of the upper null will also become more aligned with the y-axis asa is increased. The plane of the upper null’s separatrix surface will vary as a varies since the lower null’s spine must bound it.

The skeletons of the initial and equilibrium magnetic fields, for the experiments in Se- ries A, are shown in Fig. 4.1. It is clear from these plots that varying the value ofa, changes the initial magnetic field in the manner as expected from looking at the eigenvectors.

The lower row in Fig. 4.1 displays the skeleton for each experiment after the non- resistive MHD relaxation has taken place. An isosurface ofjk= 10 is also drawn on these

graphs, highlighting the twisted current layer that forms along the separator in each case. The strength of this current layer varies with a: this is discussed more in Sect. 4.3.2. Enhanced current also exists on the separatrix surfaces of the nulls for all experiments.

The formation of the strong current, which lies along the separator in all the plots shown in Fig. 4.1, indicates that the magnetic skeleton has been altered through the non-resistive MHD relaxation. The separatrix surfaces become warped through the non- resistive MHD relaxation. In Fig. 4.2a, the intersections of the initial magnetic field’s separatrix surfaces (dashed lines) are plotted in a plane perpendicular to the separator, at z = 0.5, for all five experiments discussed in Series A. For comparative purposes, on this plot, the intersection of the separatrix surfaces of the equilibrium magnetic field are plotted (solid lines), for the experiments with the lowest and highest values of a, i.e.,

a = 0.15 and a = 0.85. Here, slight differences are apparent between the positions at which the separatrix surfaces of the initial magnetic fields intersect this plane. As a is increased, the initial angle between the separatrix surfaces of the nulls enlarges slightly.

The non-resistive MHD relaxation causes the separatrix surfaces of the nulls to curve towards each other, creating cusp regions about the separator (Fig. 4.2b). In this plot the solid lines show where the equilibrium separatrix surfaces (solid lines) intersect the plane atz= 0.5 and the dashed lines highlight where the initial magnetic field’s separatrix surfaces (dashed lines) with the lowest and highest values of a intersect this plane. We see, from this plot, that the angle within the cusp regions is slightly smaller for largera. This means that the equilibrium separatrix surfaces collapse slightly more towards each other as the initial value of a is increased. Note, however, that the run times for the experiments with a = 0.75 and a = 0.85 are not as long as for the other experiments and, hence, the slight enhancements in the curvature of those separatrix surfaces could be due to them not being as relaxed. This explanation for the enhanced curvature of the separatrix surfaces is more likely since the separatrix surfaces initially appear very similar in these cuts regardless of the value ofa.

Fig. 4.2 also highlights that the separatrix surfaces, of both nulls, remain stationary on the boundaries, as required by the boundary conditions, throughout the non-resistive MHD relaxation.

In the next sections, we discuss differences in the dimensions and strength of the current layer (Sect. 4.3.2), the twist (Sect. 4.3.3) before analysing the plasma pressure and the magnetic pressure (Sect. 4.3.4) and forces (Sect. 4.3.5) in the equilibrium state of each experiment in Series A.

4.3. MHS EQUILIBRIA A: THE EFFECTS OF V AR YING P ARAMETER A 88

Figure 4.1: Skeletons of the initial (top row) and MHS equilibrium (bottom row) magnetic fields for the experiments whose initial conditions are exactly the same, save for the value of a: (a) and (f) a= 0.15, (b) and (g) a = 0.25, (c) and (h) a =0.5, (d) and (i) a = 0.75 and (e) and (j) a = 0.85. Here the lower/upper nulls are blue/red spheres with blue/red spines and pale-blue/pink separatrix surfaces, respectively. The solid pale-blue/pink lines highlight where the separatrix surfaces intersect the boundaries of the box. The separator, green line, links the null points. In the bottom row, a purple isosurface is drawn at jk = 10 in each figure.

4.3. MHS EQUILIBRIA A: THE EFFECTS OF VARYING PARAMETERA 89

Figure 4.2: Intersections of (a) the initial (dashed lines) and (b) the equilibrium (solid lines) magnetic field’s separatrix surfaces with the z = 0.5 plane for the experiments with a= 0.15 (purple), a= 0.25 (blue), a =0.5 (black), a= 0.75 (green) and a= 0.85 (orange). Over plotted on (a) are the final equilibrium positions (solid lines) of thea= 0.15 (purple) and a= 0.85 (orange) separatrix surfaces. Similarly, over plotted on (b) are the initial positions (dashed lines) of the separatrix surfaces of these two experiments. The separatrix surface of the lower null spansx =−1.0 to x= 1.0 and the separatrix surface of the upper null spans y=−1.0 to y= 1.0.