CON LA HIPÓTESIS

open sides of the separatrix surfaces, a high proportion of the particles are blue, signifying that these particles have fieldlines which lie close to the only separator, separator 1. The fieldlines attached to these particles are soon going to reconnect at separator 1. Immediately after they have reconnected, these fieldlines are quickly ejected from the separator into the closed and overlying flux domains and turn grey. Thus, in phase D3, we have shown that the reconnection which occurs at separator 1 is closing the magnetic field.

Phases D4 and D5 (Figures 4.12c and 4.12d, respectively) show a considerably increasing quantity of particles with fieldlines close to the separators. In the positive open and negative open flux domains, many of the particles near the separatrix surfaces are blue. These blue particles are about to be transported by the reconnection of their fieldlines into either the closed or overlying flux domains at separator 1. In the closed and overlying flux domains, close to separators 2 (3) a high proportion of green (red) particles have fieldlines near one of the two separatrix surfaces. These green (red) particles are moving towards separator 2 (3) and cross into either the negative (positive) open or positive (negative) reopened flux domains. In phase D3, particles which have just changed flux domain quickly turn grey, but in these phases some of these particles which have moved into an open flux domain have turned blue, indicating that their fieldlines are close to separator 1 (again). The small number of particles in the trapped negative (positive) open flux domain are mostly red (green) indicating that they have fieldlines near separator 4 (5) which are soon going to reconnect (in a closing process) at separator 4 (5). Therefore, the particle motions confirm that this experiment’s reconnection occurs at separators and that the direction of reconnection at the five separators matches our hypothesis. Reconnection at the separators is seen to have slow inflows and fast outflows which is most obvious from the coloured regions on the inflows (but not on the outflows). Finally, we see particles moving between separators in a fashion that indicates that (i) recursive reconnection is occurring, (ii) some flux may change connectivity twice (or more) in quick succession and (iii) some flux is likely to reconnect more than eight times.

4.6

Variation of the Resistivity

One question which arises from our discovery of recursive reconnection is whether the en- ergetics of our model are affected by a different evolutions of the magnetic skeleton or the total amount of reconnection which has occurred throughout the experiment. Using five experiments with five different (constant) resistivities (η = 1₈η0, 1₄η0, 1₂η0, 3₄η0 and η0, where

η0= 5×10−4) which give different evolutions of the magnetic skeleton and total amounts of reconnection (RT), as shown in Table 4.1, the following properties are then analysed:

CHAPTER 4. SEPARATOR RECONNECTION AND RECIRCULATION

(a) (b)

(c) (d)

Figure 4.13: Properties over time of the five constant resistivity runs, where η=η0 (dotted line), η = 3₄η0 (dashed line), η = ₂1η0 (dot-dashed line), η = 1₄η0 (dot-dot-dashed line) and

η = 1₈η0 (solid line). The properties shown are (a) the Poynting flux through the lower and upper boundaries, (b) the Joule dissipation, (d) the viscous heating and (d) the kinetic energy.

potential field during the evolution, 2. the Poynting flux:

Pf = 1

µ0

E_×B.

3. the Joule dissipation (or Ohmic dissipation):

Qjoule =

J2 σ .

In these five experiments, we find that the kinetic energy and viscous dissipation are small in comparison to the free magnetic energy and Joule dissipation, respectively, as shown by comparing the graphs in Figure 4.13). Hence, the kinetic energy and viscous dissipation

4.6. VARIATION OF THE RESISTIVITY

Figure 4.14: The free magnetic energy in each of the constant resistivity experiments: η=η0 (dotted line), η = 3₄η0 (dashed line), η = ₂1η0 (dot-dashed line), η = 1₄η0 (dot-dot-dashed line) andη= 1₈η0 (solid line). The grey vertical line denotes when the driving finishes. The horizontal grey line is the extension of the final free magnetic energy in the η = η0 model after the end of its run.

terms are ignored for the purposes of our present analysis.

First, we consider the free magnetic energy. The free magnetic energy in a volume V is given by Wf = Z Z Z V B2(x, t)₋B_p2(x, t) 2µ0 dV,

whereB is the magnetic field strength of our field and Bp is the magnetic field strength of the unique potential field with the same normal magnetic field components asBthrough the boundaries ofV. Graphs of the free magnetic energy are plotted for each value of η against time in Figure 4.14 and we observe the following properties. Firstly, the free magnetic energy of the five experiments at t = 0 is equal, as expected. At the end of the experiments, we also find the same free magnetic energy for all values ofη. However, Figure 4.14 shows that the free magnetic energy increases as the resistivity decreases (i.e. as the magnetic Reynolds number increases).

Figure 4.13a shows the change of the Poynting flux which adds magnetic energy to the system. Clearly, the amount of Poynting flux injected into the system is proportional to 1_η. Figure 4.13b shows the Joule dissipation which removes magnetic energy from the system (e.g. into thermal energy). Both the Poynting flux and the Joule dissipation have higher peaks when the magnetic resistivity is lowered. Such an increase in the peaks of the energy in and out of the system is is consistent with the higher maximum peak of free magnetic

CHAPTER 4. SEPARATOR RECONNECTION AND RECIRCULATION

(a) (b)

(c) (d)

Figure 4.15: The (a) total reconnection rate (RT), (b) peak Joule dissipation, (c) total Joule dissipation and (d) total Poynting flux for our five constant resistivity models with different resistivities (η). The reference value isη0 = 5×10−4. In (a), the total amount of reconnection is in terms of the total flux through the negative source.

energy which is observed for low resistivities during our experiment.

Finally, we compare the total amounts of reconnection, the peak and total Joule dissi- pation which occurs and the total Poynting flux in our experiments (Figure 4.15). We note that the total amount of reconnection falls as the resistivity increases. By examining Fig- ure 4.15a asη_→0, the maximum total amount of reconnection is likely to tend to four times the amount of flux in the negative source, and as η _{→ ∞} the minimum is likely to tend to two times the amount of flux in the negative source.∗ Similarly, the peak Joule dissipation, total Joule dissipation and total Poynting flux in our model decreases as η _{→ ∞}. By eye, the maxima of these three properties are likely to be 2.0_×10−4_{, 3.3×10}−5 _{and 4.3×10}−4_, respectively. Asη_{→ ∞}, the total and peak Joule dissipation must tend to zero, but it is not obvious as to which value the total Poynting flux tends to.

∗_{The value of the total reconnection forη =} 3

4η0 looks wrong, but this is the value given by both the

4.7. DISCUSSION