AUTOEFICACIA Y LAS TAREAS DEL DOCENTE

half-way along the separator is β_lsep/2 = 2.8

Figure 6.7: (a) The equilibrium magnetic skeleton along with a blue isosurface drawn at β = 1. Here the pale-blue/pink lines represent field lines in the separatrix surfaces of the lower/upper nulls, the blue/red lines are the spine of the lower/upper nulls and the solid pale-blue/pink lines indicate where the separatrix surfaces of the nulls intersect the boundaries. (b) Contours of the plasma beta drawn in a plane perpendicular to the separator, at z = 1.5, in the equilibrium field. The pale-blue/pink lines highlight where the separatrix surfaces intersect this cut.

6.5

Current layer formation

During the non-resistive relaxation, a strong current layer builds along the separator and enhanced current grows along the separatrix surfaces of the null points. Fig. 6.8 shows the MHS equilibrium magnetic skeleton with an isosurface ofjk = 10.0 from two different

angles. In this figure, the current is shown to be stronger along the separator away from the null points and is strong on the separatrix surfaces close to the domain boundaries.

Fig. 6.9 shows surface plots of |j| in planes perpendicular to the separator in the equilibrium state. These graphs show the strong current which lies along the separator, and is twisted along its length (discussed in more detail below), and the enhanced current which sits on the separatrix surfaces of the nulls. The value of|j|on the separatrix surfaces is shown in Fig. 6.10 normalised to the same value of |j| used in Chapt. 3 for the main experiment. We note that, unlike the high plasma-beta experiment where the value of |j|

increased going from the top to the bottom of the lower null’s separatrix surface, and vice versa for the upper null’s separatrix surface, here the current does not follow as neat a pattern. In Fig. 6.10a, which shows contours of|j|on the lower null’s separatrix surface,

6.5. CURRENT LAYER FORMATION 183

Figure 6.8: (a) and (b) show the MHS equilibrium skeleton with isosurfaces of jk =

10.0 from two different angles. The skeleton contains a positive/negative null (blue/red spheres) which have blue/red spines and pale-blue/pink separatrix surfaces. The solid pale- blue/pink lines indicate where the separatrix surfaces intersect the domain. A separator links the nulls (green line).

the current is small near the top of this null’s separatrix surface, slightly greater where it is level with the separator, and increases near the positive side of the lower boundary. Similarly for the upper null’s separatrix surface (Fig. 6.10b), the current is small near the bottom, is increased around the level of the separator on the positive boundary and has higher boundary currents at the top of the domain. Note that the value of |j| in the separator current layer is about 1.5 times greater than the maximum value on the colour bar in Fig. 6.10.

This experiment was run for a much shorter time than the high plasma-beta exper- iments and so the current has not had as long to build up along the separator. After

t = 9.34tf, multiple separators existed in the domain which linked the original two null

points together. This is an indication that the topology was changed and, hence, we ended the experiment at this point.

We define the length of the current layer as the distance between the null points along thez-axis, i.e., the length of the separator. The equilibrium current layer found in this low plasma-beta experiment is longer than the high plasma-beta current-layer length, since in the initial field the nulls are positioned further apart (but it is not as long as the initial separator in this experiment, as already discussed in Sect. 6.4, Fig. 6.5).

Fig. 6.11a shows the value ofjk, along the normalisedz-axis, in the MHS equilibrium.

The z-axis is normalised here according to the equation z∗ = L(z−zln)/lsep where L

is the initial length of the separator, zln is the z-coordinate of the lower null and lsep

is the length of the equilibrium separator. The value of jk is positive along the length

of the separator, and is negative outwith it along the z∗-axis (Fig. 6.11a), however, the profile ofjkalong the separator is different to that seen in the high plasma-beta relaxation

6.5. CURRENT LAYER FORMATION 184

Figure 6.9: Surface plots of |j| in planes perpendicular to the separator at (a) z = 0.05, (b)z= 0.5, (c) z= 0.95, (d)z= 1.5, (e) z= 2.2 and (f)z= 2.9 at t= 9.34tf.

6.5. CURRENT LAYER FORMATION 185

Figure 6.10: Contours of |j| drawn on the separatrix surfaces of the (a) lower and (b) upper nulls att= 9.34tf.

experiments discussed in Chapts. 3 and 4 wherejksmoothly increased from the lower null

to a point around z∗ = 0.4L (where L = 1) and then smoothly decreased towards the upper null. Here, the peak value of jk occurs at z∗ = 0.51L, which is just greater than

half way along the separator, and the value of the peak (jk = 35.04) is roughly six times

greater than the value of jk at the nulls (jk = 5.5 at the lower null and jk = 6.7 at the

upper null). The peak magnitude of the current along the separator is greater here than in the high plasma-beta experiments and the value of jk is slightly greater at the upper

null than at the lower null.

The value of jk is plotted in a 1D-slice through the depth (solid line) and across the

width (dashed line) of the current layer, at z = 1.5, in Fig. 6.11b. This plot shows that the current peaks at the current layer and falls off away from it, as was observed in the high plasma-beta relaxation experiments. However, the value of jk plotted through the

depth becomes very small on the edge of the current layer, then increases slightly before decreasing away from the separator current layer. This behaviour was not observed in the high plasma-beta experiments.

We calculate the width and depth of the current layer using the contour method, which is first detailed in Chapt. 3. This method involves using the last contour of |j|, in planes perpendicular to the separator, that is elliptic in shape and not X-shaped, i.e., so that the

6.5. CURRENT LAYER FORMATION 186

Figure 6.11: jk plotted (a) along the z∗-axis and (b) through the depth (solid line) and

across the width (dashed line) of the current layer in a plane perpendicular to the separator atz= 1.5 in the equilibrium state at t= 9.34tf.

Figure 6.12: The width (dashed lines) and depth (solid lines) of the equilibrium current layer found using the contour method.

area within the contour only picks up the current in the separator current layer. Fig. 6.12 shows the width (dashed lines) and depth (solid lines) of the low plasma-beta current layer found using the contour method. The width of the current layer decreases close to the nulls and bulges in the middle. This was expected from Figs. 6.8 and 6.11a in which the value ofjk appeared stronger along the separator away from the null points. The depth of

the low plasma-beta current layer also decreases at the null points and is fairly constant along the length of the separator.

As in all non-resistive relaxation experiments discussed in this thesis, the current layer formed along the separator is twisted. The angle, through which the current layer twists from the lower to the upper null, is θ= 0.6 rads. This is an average value compared to the angle through which the current layers of all the high plasma-beta experiments twist in Chapts. 3 and 4.

Next, we examine the plasma and magnetic pressure in the MHS equilibrium state and check to see if the system is in pressure balance.