Evolución en el diseño de la geometría del bloque

The formation of current singularities at line-tied 3D null points in non-resistive magnetohydrodynamics has been studied by Pontin and Craig (2005). They show how these current singularities are formed in an equivalent manner to that in two dimensions, using a frictional code with no heating term. In agreement with their study, we find that the current density spreads mainly over the fan surface, with a smaller amount also concentrating along the spine, as shown in Figure 5.15.

In our two-dimensional results from Chapter 4, we investigated the formation of a singular current at the location of the null. It was found that small residual forces remained, concentrated about the null, stretching the

Figure 5.18: Current density along the spine (black-solid), along the tilt-axis of the fan (blue-dashed) and along they-axis of the fan (orange-dashed), for the same experiment as in Figure 5.11

Figure 5.19: Magnitude of the residual forces (a) along the spine, (b) along the tilt-axis of the fan and (c) along the

y-axis of the fan.

current layer in one direction and trying to converge it to a singularity in the other. Furthermore, as the grid resolution increased, so did the forces. Also, as the system evolved, the peak current kept slowly increasing. We aim to do a similar analysis for our three-dimensional null point. In Figure 5.18, we show plots of current density along the spine, along thex-axis of the fan surface (tilt-axis), and along they-axis of the fan surface. Along the spine, a broad gradual increase of current over 0.3 length units from the null sees the current rise up from 0.7to13before it peaks at over30at the null itself. This first broad enhancement is partly associated with the current accumulation about the fan plane. After that, the current shows a spike, which reveals the formation of the singularity at the null. The region in which the current distribution along the spine coincides with the current distribution along they-axis of the fan is the region in which both the fan and spine lie concurrently, and where the singularity is to be formed.

By evaluating the residual forces, we expect to see a further indication of the formation of a singularity, but the low resolution of the experiments does not allow a conclusive result. The total forces evaluated along the spine show a sudden increase about the null, but are nearly zero at the exact location of the null itself (Figure 5.19a).

These may indicate the formation of a singularity, as studied in the two-dimensional case. However, the fan shows higher residual forces away from the null (Figure 5.19b), except in the axis of tilt of the fan, in which the forces are minima (Figure 5.19c). The reason why these forces are not sufficiently small is because they lie within the numerical error of the finite difference derivatives. The use of higher resolution runs is necessary for a firmer conclusion.

5.4.4

Changes in current density and plasma pressure

Finally, we discuss how the previous results vary when the initial current density and initial plasma pressure are varied, and compare our results to those of Pontin and Craig (2005). Pontin and Craig (2005) find a reduction in the peak current when the plasma pressure is increased.

Following the investigation of Craig and Litvinenko (2005) for the two-dimensional X-point relaxation, Pontin and Craig (2005) evaluate the scaling of the peak current density with the numerical grid size. As before, this scaling law does not make sense in our full MHD numerical experiment, as the equations do not permit the peak current to achieve a genuine singularity, but keeps slowly increasing as more time elapses.

First, from equation (5.4.4), the initial disturbance of the field (tilt of the fan) is defined by the magnitude of the initial constant current density. The higher the initial current density, the higher the angle of tilt. We find that the deformation of the spine in the final state directly depends on the steepness of the initial fan plane. The larger the initial current the greater the deformation (curvature) of the fan and the spine (Figure 5.20a). Similarly, a larger initial magnitude of the initial current density produces a steeper initial inclination and a bigger deformation of the fan plane (Figure 5.20b). The shape of the singularity along the spine for the two experiments withj0 = 0.5and 1.0is shown in Figures 5.20c and d. Here we see that the strength of the singularity increases when increasing the initial current, and a clear well defined spike is only observed for the smaller values ofj0, as the current layer gets broader for larger values of the initial current.

Changes in the initial background plasma pressure do not affect the the initial tilt of the fan surface, but they affect the final collapse of the fan and spine towards the null by varying the degree of the deformation (Figure 5.21a-b). It can be seen that a larger initial magnitude of the plasma pressure produces a smaller deformation of the fan plane and the spine line. This is not surprising since the plasma acts to reduce the effects of the initial Lorentz force, preventing a collapse in the null. The suppression of the current layer and singularity as the plasma pressure decreases is also seen in Figure 5.21c and d, where a weaker current in the fan surface and at the null is seen for a smaller plasma pressure.

Overall, the results are similar to that of the relaxation of two-dimensional magnetic X-points. The singularity becomes less pronounced if the initial current density is decreased, or the initial plasma pressure (and hence, the plasma beta) is increased.

Figure 5.20: Dependence with current density, showing (a) fan (pink) and spine (blue) in the planex = 0for

j0 = 0.5,1.0,1.5, (b) shapes of the fan surfaces, after subtracting the initial tilt, and (c) and (d), current density along the spine (black-solid) and along the tilt-axis (blue-dashed) andy-axis (orange-dashed) in the fan plane, for two of the experiments, withj0= 0.5and1.5.

Figure 5.21: Dependence with plasma pressure, showing (a) fan (pink) and spine (blue) in the planex= 0for (a)

p0 = 0.5,1.0,1.5, (b) shapes of the fan surfaces, after subtracting the initial tilt, and (c) and (d) current density along the spine (black-solid) and along the tilt-axis (blue-dashed) andy-axis (orange-dashed) in the fan plane, for the two experiments withp0= 0.5and1.5.

5.4.5

Overview

In the last set of experiments of this thesis, we have considered the dynamical evolution of three-dimensional magnetic null points with a shear-type perturbation in which the fan plane is tilted with respect to the spine, about a given axis. An initially homogeneous current density, perpendicular to the spine, pointing along thex-axis, evolves in time by collapsing the spine and the fan surface towards each other. The current density remains purely along thex-axis and it is accumulated around the final surface of the fan, and also along the spine, although with a much smaller magnitude. A large and very localised three-dimensional current layer with finite dimensions is formed about the null. It is found that this layer is wider in the direction of the tilt-axis of the fan, while it has a similar form along the spine and along they-axis of the fan, in the region in which both the spine and the fan lie concurrently.

Thex= 0plane, to which the current density vector is perpendicular, shows a very similar structure to the two-dimensional case: (1) a cusp-like enhancement in current is found outlining the fan surface and the spine, (2) pressure is enhanced in the regions inside the cusp, (3) entropy peaks at the location of the null, (4) the current density tries to become singular at the null, but a true singularity is not possible to reach numerically, so instead a pronounced spike in current is seen about which small residual forces are trying to converge the current to a singular value.

The effects of decreasing the current density or increasing the plasma pressure are, first, to lessen the collapse of the spine and the fan, and second, to decrease the strength of the singularity at the three-dimensional null point, but producing a narrower layer about the null. These results agree qualitatively with Pontin and Craig (2005), but, as in the two-dimensional case, an evaluation of the magnitude of the peak current is not of any use, as in our case, residual forces keep feeding current to the null, trying to achieve an “impossible” singularity. The field is therefore in a quasi-static equilibrium, but strictly speaking, an equilibrium is impossible to be reached using an ideal MHD evolution (this is also true in the 2D X-point collapse).

In the last chapter, after giving a brief summary of the results found in this thesis, we discuss the implications of the equilibrium states found in two-dimensional X-points and three-dimensional magnetic nulls for current sheet formation and magnetic reconnection, which, in a resistive medium, would occur around the locations of large current density accumulations.

Conclusions and Future Work

6.1

Discussion

In this thesis, the dynamical relaxation of four different hydromagnetic environments to static equilibria have been studied in detail, under the assumptions of zero resistivity and zero gravity, but taking viscous terms into consideration as mechanisms for damping velocities and heating the plasma. In each case, the initial state has been set up as a certain type of disturbance to a potential equilibrium, with no initial flows. The domain of study has been set to be closed for all the plasma and magnetic quantities. We have then analysed the electric current density accumulations at the final equilibrium states, which, in all cases, are non-force-free in nature, and thus the plasma pressure gradients are able to hold finite thick current layers at localised regions. We now summarise the results obtained in this thesis and give some conclusions for the four sets of experiments, being, the relaxation of parallel magnetic fields, of 2D magnetic X-points, and of 3D magnetic nulls with spine-aligned and fan-aligned current.

6.1.1

Results overview