RENDIMIENTO ACADÉMICO

Figure 5.28: Coloured boxes representing the volumes over which the transport of energy equations are integrated. The volumes increase according to −(0.15 +k/10) ≤ x, y ≤

(0.15 +k/10) and−0.2≤z≤1.2 shown by the colours black (k= 0), purple (k= 1), blue (k= 2), cyan (k= 3), lime (k= 4), green (k= 5), yellow (k= 6), orange (k= 7) and red (k= 8).

5.7

Transport of energy

As has been discussed previously, the separator reconnection which occurs in the system causes waves to be launched from the diffusion site, as a consequence of the sudden lack of force balance. The waves, which travel out causing changes to the magnetic field and plasma, set up flows in the system. We now study the transport of magnetic energy, internal energy and kinetic energy equations, integrated over volumes within our domain, in order to analyse what quantities these waves and flows carry with them.

We integrate the various terms in the transport of energy equations over volumes, of increasing size, within our domain. The smallest volume we integrate over, encloses the separator current layer and has dimensions −0.15 ≤x, y ≤0.15 and −0.2 ≤z ≤1.2. In total there are nine volumes, over which we integrate, with dimensions−(0.15 +k/10)≤

x, y≤(0.15 +k/10), fork = 0,1,2, ...,8 and the z range is fixed for all volumes. Hence, the largest volume we use is slightly smaller than the size of our domain. A cartoon of these volumes is shown in Fig. 5.28, where the volumes are coloured black, purple, blue, lime, green, yellow, orange and red as they increase in size. Hence, in Fig. 5.29, which shows the terms in the transport of energy equations, there are nine lines, coloured to match the different volumes.

5.7. TRANSPORT OF ENERGY 163

Figure 5.29: Quantities, plotted against time, of (a) the rate of change of the magnetic energy, (b) the Ohmic dissipation, (c) the Poynting flux, (d) the work done by the Lorentz force, the work done by the viscous force and the work done by the pressure force, (e) the rate of change of internal energy, (f) the enthalpy flux, (g) the rate of change of kinetic energy and (h) the bulk kinetic energy flux. The coloured lines represent different volumes over which these quantities have been integrated. The black dashed vertical line highlights where phase I ends and phase II begins and the black dashed horizontal line represents the zero line.

5.7. TRANSPORT OF ENERGY 164

5.7.1 Transport of magnetic energy

Eq. 5.2 is the transport of magnetic energy equation

∂ ∂t(

B2

2µ0

) =−ηj2− ∇ ·(E×B)−v·(j×B). (5.2)

This equation states that the rate of change of magnetic energy is equivalent to the negative value of the Ohmic dissipation minus the Poynting flux minus the work done by the Lorentz force.

The term on the left hand side of Eq. 5.2 is shown in Fig. 5.29a, plotted throughout the reconnection experiment, integrated over the nine volumes shown in Fig. 5.28. The negative sum of the next three graphs (Figs. 5.29b, 5.29c and 5.29d) make Fig. 5.29a in which an initial sharp drop in the rate of change of magnetic energy is observed. This drop is due to the Ohmic dissipation at this time (Fig. 5.29b) and coincides with phase I where the reconnection rate is high and a lot of Ohmic heating is occurring in the system, as was shown in Fig. 5.4. The value of the Ohmic dissipation integrated over all nine volumes is the same regardless of the size of the volume since the value of η is only non-zero in the smallest box. Hence, the integrals of the Ohmic dissipation within all larger volumes do not make any contribution to the integrated value within the black box. The Ohmic dissipation is fairly constant after t= 0.1tf which is the slow-steady second phase.

The second term, on the right hand side of value of Eq. 5.2 represents the Poynting flux. Fig. 5.29c shows that the Poynting flux is initially positive indicating that waves are travelling out through the boundaries of our volumes. As has been discussed, these waves lead to flows being set up in the system near to the original diffusion site. These flows bring Poynting flux in through the smaller volumes, over which we integrate, and eventually dominate over the value of the Poynting flux carried out of the five smallest volumes by the waves. However, in the larger volumes the Poynting flux values carried out through the volumes by the waves, become relatively large the further out they travel (see the green to red lines in Fig. 5.29c). Note, however, that the amount of Poynting flux is roughly 25 times smaller than the Ohmic dissipation which peaked in phase I.

The Lorentz force is working to try and regain force balance in the system from the moment the current in the separator current layer begins to be dissipated (positive values in Fig. 5.29d). This figure shows that the work done by the Lorentz force, which is roughly of the order of 3×10−5, is acting out through the sub-volumes, over which we have integrated. The Lorentz force is the sum of the magnetic pressure and tension forces. The magnetic tension force, which acts to straighten the field lines, is directed outwards from the diffusion site both within and outwith the cusp regions which are formed by the separatrix surfaces of the nulls. The magnetic pressure force is directed in towards the diffusion region within the cusps and outwith the cusps. Overall, these forces sum such that the work done by the Lorentz force is acting outwards away from the diffusion region. The magnitude of this term is roughly 30 times smaller than the Ohmic dissipation term.

5.7.2 Transport of internal energy

The transport of internal energy equation can be written as

∂

∂t(ρ) =ηj

5.7. TRANSPORT OF ENERGY 165

whereρ= 3p/2, since our closure equation is =p/ρ(γ−1), and γ = 5/3 and therefore

p+ρ= 5p/2. Eq. 5.3 states that the rate of change of internal energy is due to the Ohmic heating minus the enthalpy flux minus the work done by the pressure force. Fig. 5.29e shows the rate of change of internal energy, which is of the order of 10−3_{, integrated over}

all nine volumes, throughout the experiment. The initial sharp spike in this figure is due to the Ohmic heating (Fig. 5.29b) as was seen in Fig. 5.4a.

After this spike, the rate of change of internal energy decreases and becomes negative. A small travelling wave is seen moving out through all the sub-volumes which comes from the enthalpy flux term (Fig. 5.29f). This term is of the order of 5×10−4 and is 2 times smaller than the Ohmic dissipation which occurs in phase I. It remains constant in amplitude except just before it leaves the final volumes. Over a much larger area this term may become more important.

The final term which contributes to Eq. 5.3 is the work done by the pressure force (negative values in Fig. 5.29d), however, the magnitude of this term is about 30 times smaller than that of the Ohmic heating and about 15 times smaller than the enthalpy flux terms and so its contribution is small here. The work done by the pressure force is directed in through the sub-volumes over which we integrate and is acting, like the Lorentz force, to try to regain force balance in the system as soon as the reconnection begins.

5.7.3 Transport of kinetic energy

We now consider the transport of kinetic energy equation

∂ ∂t ρv2 2 =v·(j×B) + (−v· ∇p) +v·Fν− ∇ · ρv2 2 v . (5.4) This equation states that the rate of change of kinetic energy is equal to the work done by the Lorentz force plus the work done by the pressure force plus the work done by the viscous force minus the bulk kinetic energy flux.

Fig. 5.29g shows the term on the left-hand side of Eq. 5.4 integrated over all nine volumes shown in Fig. 5.28. This figure indicates that the rate of change of kinetic energy is very small (∼5×10−7) throughout the reconnection experiment. These values are so small since the work done by the Lorentz force and the work done by the pressure force are about equal in size, but are acting in opposite directions (Fig. 5.29d). Both the Lorentz and pressure forces are acting to regain force balance in the system after the waves have moved out from the diffusion site. Also, the contribution from the work done by the viscous forces (Fig. 5.29d) is very small (∼ 5×10−7) as is the bulk kinetic energy flux term (Fig. 5.29h) since the velocities in the system have small magnitudes.

5.7.4 Main terms involved in the transport of energy

There are five terms which play a significant role in the transport of energy in our high plasma-beta reconnection experiment. Ohmic heating initially causes a lot of the magnetic energy to be converted into internal energy and plays the most significant role in our experiment. This energy is then carried away from the diffusion region by the enthalpy flux, which is half the size of the Ohmic heating term and the Poynting flux, which is roughly 25 times smaller than the peak Ohmic heating. The final two important terms