Current layer dimensions

In order to determine the dimensions of the current layer, we need to define where it starts and ends. In Fig. 3.18a, the distribution of the parallel current in the equilibrium along the z∗-axis for all five experiments is plotted. The z-axis has been normalised in this figure, and in subsequent figures, for each experiment so the nulls are located at z∗ = 0 and z∗ =L(= 1) in every case. Thus, the separator always has a length of one according toz∗ = (z−zln)/lsep wherezlnis thez-coordinate of the lower null andlsepis the length of

the separator in the equilibrium field. The parallel current,jk, along thez∗-axis is positive

within the separator, but drops sharply at the nulls becoming negative in sign outside the separator. These negative values increase slightly before decreasing again away from the separator. The strong currents at the top and bottom boundaries are a result of the line-tied boundary conditions on the magnetic field which prevent the separatrix surfaces from moving. The local minima in current magnitude just outside the separator, in the experiments with the largest initial currents, suggest that the separator current layers have reverse currents at their ends. Although not commonly discussed, reverse currents have also been found associated with current layers formed at 2D null points [e.g., Titov and Priest, 1993, Bungey and Priest, 1995].

The parallel current along the z∗-axis (Fig. 3.18a) has an asymmetric profile, with a greater value as you approach the lower null from within the separator than as you approach the upper null from within the separator, in all experiments. We suspect this is due to asymmetries in the initial field and investigate this further in Chapt. 4.

Therefore, the length of the current layer is defined here as the distance between the two null points (i.e., the length of the equilibrium separator). These are also the points at which the current changes sign. This means we do not include the reverse current regions when considering the length of the separator. The equilibrium separator lengthens as the

zpositions of the lower and upper nulls move further apart asjsepis increased (Fig. 3.18b).

If the initial current tended to zero, the null points would not move apart and the separator would maintain length L. The growth of the current layer indicates that as a current layer is formed the separator itself can be lengthened significantly. The amount

3.3. THE EFFECTS OF VARYING JSEP 74

Figure 3.19: Plot of (a) |j|through the depth (solid lines) and across the width (dashed lines) of the current layer, in the cut at z = 0.5 across the separator. The asterisks and diamonds represent the values of the current contour used to determine the depth and the width of the current layer at this cut (for the contour method) and the crosses and triangles (for the FWHM method), respectively. The colours represent experiments with initial uniform currentjsep = 0.75 (black),jsep= 1.0 (blue),jsep= 1.25 (green),jsep = 1.5

(orange) andjsep= 1.75 (red). (b) Outline of the width (dashed) and depth (solid) of the

current layer using the contour method (pink lines) and the FWHM method (blue lines) for the experiment withjsep = 1.5.

by which it extends will depend not just on the current accumulated, but also on the properties of the plasma and the velocity flows within the system.

Defining the width and depth of the current layer is not trivial since the current gradually decreases rather than abruptly stops (Fig. 3.19a). In Fig. 3.19a, 1D slices of |j|

are plotted in thez= 0.5 plane, through the depth (solid) and across the width (dashed) of the current layer for all five experiments. The 1D slices of |j| through the depth of the current layer show significantly enhanced currents forming a narrow peak about the separator. Elsewhere the current is small.

We consider two approaches for measuring the depth and width of the current layer; counting the current down to the saddle point and using the full width at half maximum. The first method (which we shall call the contour method) involves examining contours of

|j|, in cuts perpendicular to the separator across the current layer. We then plot a contour in each cut at a value of |j| which only outlines the current layer and not the enhanced current along the separatrix surfaces. This contour looks elliptical in nature and we define the largest diameter of this contour to be the current-layer width, whilst the smallest diameter is defined as the current-layer depth. In other words, we count only the current down to the saddle point of|j|to pick out the current layer. Once the level of the contour has been found for all perpendicular cuts through the separator, the width and depth of the current layer, along the length of the separator, can be determined. The values of the current layer depth and width, atz= 0.5, found using this method are highlighted in Fig. 3.19a by the asterisks (depth) and diamonds (width).

The second method uses the full width at half maximum (FWHM) of the current plotted in Fig. 3.19a. Fig. 3.19a displays the values of the current layer depth and width,

3.3. THE EFFECTS OF VARYING JSEP 75

in the plane atz= 0.5 using this method, plotted as crosses (depth) and triangles (width) but, note, the value of the maximum varies in each cut. Fig. 3.19b compares the current layer widths and depths, found using the contour method and the FWHM method for the experiment where jsep = 1.5. We have chosen the contour method as the preferred

method to proceed with as it was found that the contour level, using the contour method, in differentz-cuts varied less than when using the FWHM method and the contour method accounts for more current in the current layer than the FWHM method.

Figure 3.20: Outline of the (a) depth and (b) width of the current layer defined using the contour method. (c) Depth (diamonds) and width (asterisks) of the current layer defined using the contour method at z = 0.5. Crosses and triangles represent the width and depth, respectively, found using the FWHM method atz= 0.5. The blue/black lines join the widths and depths found using the contour/FWHM methods. The colours represent the results for the experiments with initial current jsep = 0.75 (black), jsep = 1.0 (blue),

jsep= 1.25 (green), jsep= 1.5 (orange) andjsep= 1.75 (red).

In Figs. 3.20a and 3.20b, the edges of the current layer through the depth and across the width determined using the contour method are plotted, against z, for the different experiments. The depths seem to remain pretty constant along the length of the current layer. The widths of the current layers, however, appear to decrease slightly near the nulls at the ends of the separator, for the experiments with higherjsep. The width of the

current layer clearly increases withjsep, as does the depth, but by a smaller amount.

In order to determine how the width and depth of the current layers depend on the initial current, we plot the depth and width, as determined in thez = 0.5 plane, against

3.3. THE EFFECTS OF VARYING JSEP 76

jsep(Fig. 3.20c). The values of the width and the depth calculated using both the contour

and the FWHM method are plotted here. While the FWHM method does not pick out a lot of the current seen in the sharp peak near the current layer it does, however, indicate that, the higher the initial currentjsep is, the closer the equilibrium current layer appears

to be to a singularity, as the depth of the current layer is seen to decrease with increasing initial current.

Using the contour method of counting the current down to the saddle point, the depth of the current layer grows with initial current jsep (Fig. 3.20c). Similar behaviour is

observed for |j| across the width of the current layer, but for all finite initial currents

jsep, the width of the current layer is much greater than the depth. The rate of increase

in width grows with jsep. The width of the current layer is found to be around 5.7jsep

bigger than the depth for jsep = 0.75, 6.3jsep bigger for jsep = 1.0, 7.2jsep bigger for

jsep = 1.25, around 8.3jsep bigger for jsep = 1.5 and around 8.7jsep times forjsep = 1.75

in the plane z = 0.5. Note, that the values of width and depth plotted in Fig. 3.20c for the experiment with initial currentjsep = 1.75, are plotted at an earlier time than for the

other experiments so this experiment is not quite as relaxed as the others.