FACTORES QUE CONDICIONAL EL ESTRÉS.

Att= 0tf, not only is there a non-zero Lorentz force acting on the model (as was seen in

the experiments in Chapts. 3 and 4), but due to the initial form of the plasma pressure, a non-zero pressure force is also acting on the model. The strength and direction of this total force (equivalent to the Lorentz force plus the plasma pressure force) is shown in a plane perpendicular to the separator, at z = 1.5, in Fig. 6.3a. The total force acts in towards the separatrix surfaces in this plane, indicating that as soon as the experiment begins, the separatrix surfaces will fold towards each other.

6.2. LOW PLASMA-BETA MAGNETIC FIELD AND PLASMA PROPERTIES 177

Figure 6.2: (a) The initial magnetic field skeleton which contains a positive/negative null (hidden by isosurfaces) with blue/red spines and pale-blue/pink separatrix surfaces. The solid pale-blue/pink lines indicate where the separatrix surfaces intersect the boundaries. A separator links the nulls (green line). Also, blue isosurfaces are drawn here atβ = 1.0. (b) Contours of the plasma beta drawn in a plane perpendicular to the separator half-way along its length atz = 1.5. The pale-blue/pink lines show where the separatrix surfaces of the nulls intersect these cuts. These intersections are plotted on top of thick white lines so they are visible here.

Figs. 6.3b and 6.3c show contours of the total force in planes perpendicular to the y

and x-planes, at y = 0 and x = 0, respectively. In both of these planes the total force is shown to act inwards, from the boundaries, towards the z-axis, except at the null points where, due to the nature of the pressure profile, the total force is directed outwards in the

x and y-directions for Figs. 6.3b and 6.3c, respectively. The stronger pressure here acts against the collapsing of the separatrix surfaces towards each other during the relaxation. The low plasma-beta single-separator model is allowed to relax non-resistively over time using the Lare3d code. We analyse the results found throughout this relaxation in the following sections, starting with a discussion of the energetics of the experiment in Sect. 6.3. Next, we analyse how the magnetic field and plasma evolve (Sect. 6.4) before detailing the properties of the current layer which forms through the relaxation (Sect. 6.5). We then analyse the plasma pressure and the magnetic pressure in the equilibrium state

6.3. ENERGETICS 178

Figure 6.3: Contours of the total force in a plane (a) perpendicular to the separator at

z = 1.5, (b) perpendicular to the y-axis at y = 0 and (c) perpendicular to the x-axis at

x= 0. The orange arrows show the direction and strength of the (a) x and y, (b) x and

z and (c)y and z components of the total force in these planes, respectively.

(Sect. 6.6) and look at the properties of residual forces which exist in the MHS equilibrium (Sect. 6.7). We discuss the growth rate of the current in Sect. 6.8 before detailing how a low mean plasma-beta value can be achieved by using the MHS equilibrium field (Sect. 6.9). Our findings are summarised in Sect. 6.10.

6.3

Energetics

The initial magnetic field is not in force balance and so, as soon as the experiment begins, the separatrix surfaces collapse towards each other. During the experiment, overall, the magnetic energy is converted into kinetic energy, which in turn is converted into internal

6.3. ENERGETICS 179

energy (Fig. 6.4). However, the path that the energies take to being constant is different to that observed in the high plasma-beta relaxation experiments discussed in Chapts. 3 and 4. The system evolves to what appears to be a MHS equilibrium byt= 9.34tf which

is the end point of the experiment. By this time there have been no significant changes to the energies for almost 7tf.

Figure 6.4: Plot of the kinetic (green), magnetic (blue), internal (orange) and total (black solid) energies along with the viscous (red dashed) and adiabatic (cyan dashed) heating terms. These energies have been scaled such that the total energy is 1.0, the initial/final magnetic energy values match the final/initial internal energy values and the adiabatic heating starts at the same value as the internal energy, with the viscous heating term starting from the final value of the adiabatic heating. The dotted and dashed black lines highlight the start and end points of the heating terms.

The curves of the energy and heating terms, in Fig. 6.4, have been shifted on they-axis, for representational purposes, such that the internal energy starts/finishes at the values at which the magnetic energy finishes/starts. This is to highlight that all the magnetic energy, to within numerical error, is converted into internal energy. The magnetic energy starts to decrease as the experiment begins and the kinetic energy increases, however, rather than an increase in internal energy following this, the internal energy decreases due to the adiabatic cooling and appears to go into both kinetic and magnetic energy at around t = 0.15tf. After this time the magnetic energy increases slightly, due to the

adiabatic cooling before decreasing as it is converted into kinetic and internal energy. The internal energy begins to increase aftert= 0.15tf due to the conversion of kinetic energy

to internal energy via viscous heating.

This behaviour, from the start of the experiment to t = 0.15tf, occurs since there

is a rapid expansion of the plasma in the system as the experiment begins due to the pressure gradients about the nulls. This expansion is adiabatic in nature causing there to be cooling in the system, evident from the drop in the adiabatic term, plotted in Fig. 6.4, near the start of the experiment. The viscous heating term increases throughout the experiment, indicating that waves are being damped. The existence of waves is evident from the oscillations of the curves in Fig. 6.4.