Current density layer

We now look closely at the dimensions of the current layer of the sample experiment and evaluate the nature of its finite width. Figure 4.11 shows cuts along the length (horizontal cut) and width (vertical cut) of the current layer at the location of the null, showing a length and width of the current layer which are respectively of around 23 and 15 points (around 0.02 and 0.005 length units). These may indicate that the current layer really has finite dimensions and are not a result of the resolution of the numerical experiment, as suggested in Rast¨atter et al. (1994). In order to check this, in Figure 4.11 we overplot the results from the same experiment, run with a resolution of512×1024 (half the original resolution). As can be seen, the dimensions of the current layer coincide for both experiments, i.e. the finite width of the current layer is not a resolution effect, but a real characteristic of the equilibrium. These results contradict the ones from Craig and Litvinenko (2005), for which the dimensions of the current layer are decreased when increasing the resolution.

In Figure 4.12, we show vertical cuts of the current density across the central current layer, for six different experiments with the same squashing,h= 0.7, but with different initial pressures. The width of the central layer decreases for smaller plasma pressures, but remains finite. In Figure 4.13, we show horizontal cuts of the current density along the central current layer, for the same cases as in Figure 4.12. As the pressure is decreased, the length of the central current layer extends further, and the current density becomes more concentrated, developing a higher peak. The same behaviour is observed if the initial plasma pressure is held fixed, and the height of the box is systematically decreased (i.e. the squashing is increased). This means that decreasing the initial plasma pressure has a similar effect as increasing the initial current density, as the action of both is to make the Lorentz force dominate over the pressure force.

When the initial plasma pressure is small (e.g. in Figure 4.13f), the current layer has a length that is many times longer that its width. We consider whether the current layer is approaching the form found in Green’s current sheet

Figure 4.12: Plots of electric current density across the width of the central current layer, for six different experi- ments, withh= 0.7, but different initial plasma pressures.

Figure 4.13: Plots of electric current density along the length of the main current sheet, for six different experi- ments, withh= 0.7, but with different initial plasma pressures.

Figure 4.14: The six plots in Figure 4.13 are overplotted for comparison. The dimensions of the Green’s potential solution are given in dotted lines. Figure (b) is a zoom of (a) over a smaller range of current densities. In (b), the initial current density is overplotted (dashed).

solution. This is checked by comparing these plots with the correspondant length of the Green’s current sheet (Figure 4.14).

To derive Green’s expression, complex variable notation is used to simplify the discontinuity in the magnetic field as cuts in the complex plane,Z=x+iy. The magnetic field around a potential current sheet is described as

By+iBx=

p

Z2_−a2₌p_x2_−y2_{+ 2ixy−a}2_,

where2ais the length of the current sheet. Following the derivations in Bungey and Priest (1995), the analytical profile of the current density along Green’s potential thin current sheet is given by the magnetic field discontinuity asjz= 2Bx(x, y= 0), and can be calculated from the expression above as

jz= 2Bx(x, y= 0) = 2

p

a2_−x2_{. (4.4.4)}

Integrating equation (4.4.4) along the length of the sheet, we get the total current in the sheet, as

jT =a2π . (4.4.5)

Now, current density conservation implies that the Green’s sheet associated with our equilibrium distribution should have a total current density equal to the total current density in the initial field, hence, the half-length of Green’s current sheet is directly related to the initial constant current distribution. Looking at the results in Bungey and Priest (1995), it can be seen that our normalization requires a factor of 1/4in front of this length, hence, obtaining a= 1 4 r j0 π , (4.4.6)

which for our sample experiment gives the value a ≈ 0.144. Note, that equation (4.4.4) represents a singular current sheet containing the whole current in the domain, so in a hypothetical case of a numerical Green’s state,

this would have to be compared with the integrated current density over the width of one resolution element. What we do for our experiments is compare our distributions with the lengtha.

In Figure 4.14, the six horizontal cuts of Figure 4.13 are overplotted, and the dimensions of Green’s potential solution are marked. All the curves cross at the same points onxandy, namely(±a, j0), corresponding to the initial value of the current density iny, and the two ends of Green’s current sheet inx. The main conclusion that may be extracted from these plots is that the field is in all cases very far from the potential solution, although the fact that all curves cross at the ends of Green’s potential sheet seems to imply that Green’s solution might be achieved (as far as we can get with the resolution) in the limitp0→0.

Following a systematic study, we find that the dependence of the equilibrium distributions with the initial quantities differs from one experiment to another, and is therefore determined by the initial plasma pressure and current density of the system. This is studied in detail in Section 4.5.