We start by considering the temporal evolution of the magnitude of the current density in Cases 8 and 9 before discussing the structure of the field lines. Figure 3.20shows the cross section of current density magnitude of Case 8 (left) and Case 9 (right). Due to the initial perturbation, a kink instability creates an initial current sheet (red colour in the contour plot) within the right- hand side thread at timet= 60. It matches the time at which the energy profiles begin to change for both cases. The current sheet becomes stronger att = 80for both Cases 8 and 9, with two observable vortex patterns near the thread boundary. This suggests that magnetic reconnection is likely to occur and will be discussed below. The current within the right-hand thread begins to fragment and many small-scale current structures are seen at a later time. This has been discussed by various authors (e.g. Hood et al., 2009) and seems to be a key part of the Taylor relaxation process.
The second thread is then destabilised by the first thread and this is seen in the current density plots as the formation of a second current sheet. For Case 8 this occurs att= 160and for Case 9 att= 130. By the end of each simulation,t= 300, the current has fragmented and spread across in each thread. For Case 8, the regions of small scale current remain distinct and are separated by a region of essentially potential magnetic field. However, in Case 9, there is a clear indication that the two threads have combined to form a single larger structure in Figure3.20(bottom right).
We then track the evolution of the magnetic field lines for Case 8. The field lines around the centre of each thread are traced from one photospheric end to the opposite as shown in Figure3.21
for Case 8. These are coloured red and yellow for the left-hand thread and blue and green for the right-hand thread. If there is no magnetic reconnection, then the red/yellow and blue/green field
Figure 3.20: The contour plots of the magnitude of current density, j(x, y,0), as a function of
(x, y)andz= 0at times. The figures on the left and right are for Cases 8 and 9 respectively. The colour scale for the current density goes from 0 (white) to 5 (red).
lines will lie on top of each other. If there is a reconnection, then the ends of the field lines will not locate at the original footpoints. At timet= 60, a helical structure can be seen on the right-hand side thread, as the kink instability is excited. The field lines are seen to unwind or straighten. Reconnection has occurred in Figure3.21(b) att = 80. In particular, the green field lines start from the thread axis at the far end of the right-hand thread. However, these field lines completely encircle the thread axis at the near end. The green and blue field lines do not follow the same paths indicates that magnetic reconnection has occurred. Once the second instability is triggered, the left-hand thread follows a similar evolution. By the end of the simulation, the field lines are
Figure 3.21: Case 8: the field line plots at time: (a)t = 60(top left),(b)t = 80(top right),(c) t = 160 (bottom left) and(d)t = 300(bottom right). The yellow and red field lines are drawn from(x, y, z) = (−2,0,10)and(x, y, z) = (−2,0,−10)while the blue and green field lines are drawn from(x, y, z) = (2,0,10)and(x, y, z) = (2,0,−10).
Figure 3.22: The field line plots for Case 9 at time:t= 150(left) andt= 300(right). The yellow and red field lines are drawn from(x, y, z) = (−2,0,10)and(x, y, z) = (−2,0,−10)while the blue and green field lines are drawn from(x, y, z) = (0,0,10)and(x, y, z) = (0,0,−10).
Figure 3.23:t= 300: The contour plots of current density,j(x, y,0), for Cases 10-13 with colour scale defined from 0 (white) to 5 (red).
Figure 3.24:t= 300: plots of the magnitude of current density,j(x,0,0), for four cases. The black and red curves in(a)represent Cases 8 and 12, and(b)Cases 9 and 13.
nearly untwisted and the two threads remain completely distinct.
Figure3.22shows the field line plots for Case 9 at time t = 150 (left) and t = 300 (right). What is clearly shown is that various coloured field lines are now wrapping around each other, forming one weakly twisted magnetic loop. It will be interesting to see if future simulations can
Figure 3.25: t= 300: The field line plots for Cases 10 to 13 while the field lines are drawn from the respective footpoints.
determine the maximum distances apart where the individual threads can still merge into one loop. The contour plots of current density and field line plots of Cases 10 to 13 are also shown in Figures3.23and3.25for comparison.
The final stages of the magnitude of the current densities,j(x,0,0), for Cases 10 to 13 are also presented in Figure 3.24. The threads on the left are ideally stable to ideal MHD disturbances as λ = 1.4. i.e. the current density profiles should not change if there is no external force to trigger the kink instability in the stable thread. The red curve for Case 12 in Figure3.24(a) shows the result we expected. Very little happens to the left-hand side thread as it is not affected by the unstable thread located 2 units away on its right-hand side. However, when the threads are moved together, the stable thread is then driven unstable, as shown in Figure3.24(b) (red). The initial smooth current density profile of the left-hand side thread is disrupted everywhere across
−3< x <1. There is also a maximum peak of current of2.68atx=−1.25, which is very close to where the threads are touching each other.
Figure 3.26: Temperature plots,T(x,0,0), as a function ofxaty=z= 0for Case 8 (black) and Case 12 (red). The times are(a)t= 60,(b)80,(c)160 and(d)300.