Singular current

In Section 4.4.3, we evaluated the plasma pressure and current density of the final state. We now check whether the current accumulation at the location of the null is held in a true equilibrium by evaluating force balance along and across the current layer. Figure 4.15 shows plots of the different forces along and across the current layer, namely, plasma pressure force,₋∇_p, magnetic force,j_×B, and total force,j_×B₋∇_p. At first sight (Figure 4.15a and b), the forces seem to be balanced, and the field seems to be in equilibrium. However, when we look closely about the origin (Figure 4.15c and d), there is a residual non-zero total force which appears to be trying to stretch the null in the horizontal direction, pushing from the top and bottom, and pulling from the sides. These forces could either be a result of a small amount of reconnection due to numerical diffusion, or may be the result of the current sheet trying to tend towards a singularity. If the cause is reconnection, then the amplitude of the forces at the same time of the relaxation should increase as the grid-cell size is reduced. If on the other hand, the forces are a result of the system attempting to form a singularity, they will decrease as the grid-cell size is decreased.

We have run the same sample experiment withh= 0.7andp0= 0.375for three different resolutions, namely 256×512,512×1024and1024×2048. In Figure 4.16, we show those residual forces for the same experiment after the same time has elapsed. The amplitude of the forces is higher the better the resolution is, implying that the field is trying to converge to a singularity, and the higher the resolution, the closer the field is to achieving the singularity, and so, the bigger the forces around the current layer are. Note, that the length over which these forces extend is roughly the same for the three resolutions.

Furthermore, the peak current appears to be slowly increasing in time, even when the velocities are essentially zero everywhere in the domain (Figure 4.17). This is the last evidence of a singularity being formed, and, again, represents a difference with the work in Craig and Litvinenko (2005), in which they present scaling laws for the peak current, which for our experiments, is not well defined.

4.4.6

Overview

We have presented evidence that the field has achieved an equilibrium everywhere save at the null point, where the field is trying to converge to a singularity which is different in nature to the ones found by others in the force-free cases when using relaxation codes as opposed to a full MHD code. However, this state is impossible to reach numerically, because of the resolution constraint. Nonetheless, the forces are sufficiently small for us to consider

Figure 4.15: Pressure gradient force (dashed), magnetic Lorentz force (dashed-dot) and total force (solid), along (a) horizontal and (b) vertical cuts through the X-point for the sample experiment as shown in Fig. 4.4. The total force very close to the origin is plotted against (c)xand (d)y.

Figure 4.16: The total forces along the length of the current sheet after the same elapsed time for our sample experiment shown in Fig. 4.4, but run using different grid resolutions.

Figure 4.17: Magnitude of the electric current density at the location of the null, as a function of time, for the same experiment as shown in Fig. 4.4.

this state to be a quasi-static state, which can be understood as a magnetohydrostatic equilibrium.

An important result is that the form of the functionsp(Az)andjz(Az), which define Grad-Shafranov’s con-

dition, are different for each of our experiments. That is, the final equilibrium directly depends on the initial conditions of the experiments, i.e. on the initial plasma pressure and the initial current density. Also, the plots are not symmetric with respect toAz = 0, showing that the system approaches the singularity in a different way for

positive and negative values ofAz, i.e. inside and outside the cusp, respectively.

In comparison to the study of Craig and Litvinenko (2005), they use a frictional relaxation scheme with a fictitious mechanism for damping velocities, while our MHD numerical experiments involve a physical viscous term which is associated with a heating term, which heats the plasma, taking energy from the magnetic field. This affects in various ways to the final equilibrium state. First, they find a current layer about the null whose area is decreased when increasing the resolution. However, we suggest here that the system may achieve a state with a well defined and finite width and length of the current layer. It may happen that a non-negligible heating around the null point enables a larger finite width to be held. Also, our peak current density, at the origin, is not able to achieve a stable value, due to the presence of residual forces about the null that try to collapse the field towards a singularity, even if the field has achieved a good equilibrium everywhere else. These forces continue feeding the singularity if the simulation is run for longer, and hence, the strength of the singularity (as studied by Craig and Litvinenko, 2005) is not a good parameter to evaluate. Instead, we will try to give a qualitative description of the field around the null point, and see how this depends on the initial quantities of our experiments.