Energy evolutions and maximum temperature

We begin our analysis by studying the behaviour of the three different energy terms, integrated over the computational volume. We then investigate how the maximum temperature within the computational domain is changed under the effects of thermal conduction and optically thin radi- ation.

Figure 4.1 shows the time evolution of the volume integrated magnetic, kinetic and internal energy for Cases 14 to 17. The solid curves in black are for Case 14, which show the evolution without thermal conduction and optically thin radiation effect. The dashed curves in blue are for Case 15 with thermal conduction added, while the dotted-dashed curves in red are for Case 16 which has only the optically thin radiation added. The double-dotted-dashed curves in purple are for Case 17, which show the evolution with both thermal conduction and optionally thin radiation included. All cases start with the same initial temperature of2×104 K and evolve in response to the non-linear development of the kink instability. The initial stage (up to about 65 seconds) is

Figure 4.1: The volume integrated energies as function of time (in real units) for (a) magnetic energy,(b)kinetic energy and(c)internal energy. In(d), the maximum temperature is shown as a function of time. Case 14 (black solidcurves) is the reference case which has no thermal conduc- tion and no optically thin radiation included, Case 15 (blue dashedcurves) includes only thermal conduction, Case 16 (red dotted-dashedcurves) includes only optical radiation and Case 17 (pur- ple double-dotted-dashedcurves) includes both. All of these cases have an initial temperature of

2×104K.

the development of a helical current sheet (e.g. Figure3.8), as the most unstable mode of the kink instability grows from a given initial perturbation. The initial perturbation is very small, allowing the instability to develop naturally. Hence, the initial stages behave as an ideal plasma and the magnetic energy remains unchanged. As a consequence, the maximum current also remains below the critical value (jcrit = 5), so that the resistivity remains zero and therefore the initial energy

remains at its initial value. There is no heat generated during this stage.

Around t = 50 seconds, the maximum temperature begins to rise. It is only visible in the logarithm plot of the maximum temperature in Figure4.1(d). This increase must be localised and is simply due to the compression of the plasma, as a current sheet is beginning to form. Att= 65

seconds, the volume integrated magnetic energy begins to decrease. During this stage, resistivity becomes important, as the critical current is reached. Magnetic reconnection develops, untwists

the magnetic field lines and releases magnetic energy. The magnetic energy released will heats the plasma within the loops through both ohmic heating and viscous heating in shocks. Since the initial temperature is low, slow mode shocks are easily formed. The rapid loss of magnetic energy is the same for all cases and lasts untilt= 90seconds. This comes towards the end of the rapid release of magnetic energy and the evolution is independent of the additional terms in the internal energy equation. The evolution is solely governed by the magnetic forces. Aftert= 90

seconds, the subsequent release of magnetic energy becomes significantly slower, as the field relaxes towards a lower energy state through many small-scale reconnection events. A difference can now be observed between the cases: the magnetic energy in Cases 14 and 16 (i.e. without thermal conduction) is nearly identical during the simulation, indicating that radiation effect does not have a substantial influence on the release of magnetic energy in our simulation. The reason for this is the kinked instability is due to Lorentz force and plasma β is still small (less then

0.05), so the pressure of plasma does not change the evolution of instability. Meanwhile, the loss of magnetic energy increases slightly for90 < t < 200in Cases 15 and 17 (i.e. with thermal conduction included). By the end of the simulation, the loss of magnetic energy in each case is approximately the same.

The kinetic energy curves for Cases 14 to 17 all follow the same general evolution (see Fig- ure 4.1(b)). At time t = 65 seconds, the curves begin to rise rapidly as the magnetic energy decreases. There are two peaks around t = 73 and 90 seconds, with a possible third peak at

t= 94, before the kinetic energy decreases towards zero as the field relaxes towards its final state. There are slight differences again with the cases in which thermal conduction is included. For example, in Cases 15 and 17, the kinetic energy has slightly narrower and lower peaks att= 73

and90 seconds, and is noticeably lower aftert = 94, suggesting that thermal conduction does influence the magnetic force and the plasma motion.

The volume integrated internal energy shows the largest variation of the four cases studied. As we can see in Figure 4.1(c), all the curves match each other until t = 94. When optically thin radiation is included (i.e Case 16), only a small amount of internal energy is reduced compared to the reference case (i.e. Case 14), but when thermal conduction (i.e. Cases 15 and 17) is included there is a significant reduction in the internal energy. The high temperature is quickly conducted away from localised hot spots in the volume.

The variation of the maximum temperature as a function of time is shown in Figure4.1(d) . Due to the fact that radiation is most efficient at transition region temperatures, it has no significant influence on the maximum temperature until well after the main release of energy. On the other hand, it is clear that thermal conduction reduces the maximum temperature to around 107 _{K, as}

Figure 4.2: The volume integrated energies as function of time (in dimensionless unit) for (a)

magnetic energy,(b)kinetic energy,(c)internal energy. (d)the maximum temperature over time. All cases have an initial temperature of2×104K and are under the influence of thermal conduction and optically thin radiation. Here, Case 17 (black solidcurves) is the reference case, while Case 18 (blue dashed curves) has the initial density enhanced by 2 and Case 19 (red dotted-dashed

curves) has the initial density enhanced by 3.