Dependence with initial quantities

The process can be repeated for all the numerical experiments, finding a dependence with the initial pressure and initial current density. Figure 4.21 shows the dependence of the parameters(mi, ni, mo, no)with initial pressure,

for the different values ofh, obtained using fits like the ones in Figures 4.19 and 4.20 for every single experiment. The first conclusion that we directly obtain from Figure 4.21 is that there exists a clear functionality of the four coefficients with the initial values of the pressure and current density, i.e with the plasma beta and the initial perturbation. Both mi andmo increase as the initial pressure increases, but ni andno decreases as the initial

pressure increases. The second is that the solutions for inside and outside the cusp are different, as was already assumed in Vekstein and Priest (1993).

The pressureless limit case gives some hint about the results shown in Figure 4.21. When the plasma pressure tends to zero, then one would expect the system to approach the potential case where current density is zero everywhere (except in a thin current sheet where it becomes singular), hence, the coefficientsmiandmoshould go

Figure 4.22: Here, we show the same plots as in Figure 4.21, with fits to the curves to the expressions (4.5.6) and (4.5.7), in red, for the coefficientsmiandmo, and blue for the exponentsniandno.

to zero whenp→0. With that consideration, we can find a good fit to the plots of Figure 4.21, using exponential functions, as follows,

m=−A(e−Bp0

−1), (4.5.6)

n=C(e−Dp0_{−1) +E , (4.5.7)}

where(A, B, C, D, E)are the parameters for a non-linear fitting. Figure 4.22 shows the fits of equations (4.5.6) and (4.5.7) to the plots in Figure 4.21. The non-linear fits have been done by using the Levenberg-Marquardt method for non-linear modeling, described in Press et al. (1992), “Numerical Recipes”, Chapter 15. The parame- ters after the fits are summarised in Table 4.1. No more conclusions can be obtained from the data, apart from the fact that these parameters preserve monotonicity.

In the limitp → ∞, i.e. the plasma dominates over the magnetic field, the coefficientsmi/o tend to Ai/o,

and the exponentsni/otend toEi/o−Ci/o. These are summarised in Table 4.2. As an ambitious observation,

h j0 Ai Bi Ci Di Ei Ao Bo Co Do Eo

0.9 0.23 0.112 3.786 0.432 5.848 0.596 0.109 3.971 0.370 5.501 0.540 0.8 0.56 0.314 2.669 0.339 3.550 0.470 0.313 2.341 0.417 3.893 0.558 0.7 1.04 0.570 2.392 0.306 2.764 0.434 0.689 1.312 0.532 3.772 0.685 0.6 1.77 0.860 2.366 0.273 1.420 0.370 1.271 0.829 0.575 3.208 0.748 Table 4.1: Parameters(A, B, C, D, E)as functions of the height of the boxh, or the initial current densityj0. The subscriptsi/orefer to inside/outside the cusp respectively.

h j0 Ai Ei−Ci Ao Eo−Co

0.9 0.23 0.112 0.164 0.109 0.170 0.8 0.56 0.314 0.131 0.313 0.141 0.7 1.04 0.570 0.128 0.689 0.153 0.6 1.77 0.860 0.097 1.271 0.173

Table 4.2: Limits forp→ ∞as functions of the height of the boxh, or the initial current densityj0.

exponentsni(andno) in the limitp→ ∞, would then not depend onh. This is, when the plasma pressure is very

big, the final equilibrium would not depend on the squashing, i.e. on the electromagnetic perturbation.

4.5.3

Overview

We have studied in close detail the same problem as Craig and Litvinenko (2005), making use of the full set of MHD equations, and we have found that our final equilibrium differs from their result in some aspects. Using the approach given by Vekstein and Priest (1993), we have given a qualitative description of the final equilibrium states by looking for fits to the equation jz = F(Az). Even if this is only a qualitative analysis, it describes a

fair approximation of the behaviour of the final equilibrium as the values of the initial plasma pressure and current density are varied. These two-dimensional contexts are of high relevance for systems with translational or rota- tional symmetries, and their study is useful for some astrophysical environments which can be well approximated by these properties of symmetry.

In the next chapter, we evaluate current accumulations in three-dimensional equilibria which contain 3D mag- netic null points. The characteristics of these environments are going to be completely different to the two- dimensional case, and the dynamical evolutions are less restrictive in the sense that the plasma has freedom to move in all three spatial directions. Hence, the approach to the problem will have to be different.

Relaxation of 3D Magnetic Null Points

5.1

Introduction

Three-dimensional magnetic null points have been studied in detail within the last decade in the main context of three dimensional magnetic reconnection. Their importance for magnetic energy release in solar and magneto- spheric environments have been observationally established by many authors, for example in solar flares (Fletcher et al., 2001), in solar active regions (Ugarte-Urra et al., 2007) or at the Earth’s magnetotail (Xiao et al., 2006). However, a complete understanding of the formation of a current sheet through the collapse of a three-dimensional magnetic null point is still to be achieved, either mathematically or phenomenologically.

The processes of reconnection in three dimensions are significantly different to and much more complex than those in two-dimensions at X-type null points (e.g Hesse and Schindler, 1988; Priest et al., 2003). In general, in three-dimensions, magnetic reconnection can occur either at nulls or in the absence of them, and does not involve one-to-one breaking and rejoining of pairs of field lines, as in two-dimensions. A classification of the reconnection regimes at three-dimensional magnetic null points is made by Priest and Pontin (2009). The nature of the reconnection that takes place around a three-dimensional null depends directly on the flows and boundary conditions that are responsible for the reconnection (Figure 5.1). Below, we discuss the different reconnection regimes about 3D nulls, assuming initially a potential radial null, where the spine is perpendicular to the fan, and field lines in the fan plane extend radially from the null (Parnell et al., 1996).

1) A rotation of the fan plane about the spine drives a twist of the field lines around the spine (red arrows in Figure 5.1). An electric current density builds along and near the spine, in the direction of it. Reconnection of field lines may take place in that region, producing a slippage of the field lines in a counter-rotational direction to the twist, which dissipates the current density. This is called torsional spine reconnection. It does not involve flow across the fan or the spine, and hence, the global topology of the field remains unchanged. Note, that the reconnection does not take place at the location of the null. Models of torsional magnetic reconnection are given by Pontin et al. (2004) and Wyper and Jain (2010).

2) A rotation of the field lines about the spine, in different directions above and below the fan, creates an electric current in the fan plane that points in the direction of the spine but has different sign above and below the fan (yellow arrows in Figure 5.1). Reconnection of field lines can take place in the regions near the fan by a rotational slippage of the field lines there in opposite directions above and below the fan, so as to dissipate the

z y x (a) (b) (c) (d)

(2) Direction of the induced current density vector

y x z (1) Boundary disturbance (0, 0, j )0 (0, 0, j )0 ( , 0, 0)j0 ( , 0, 0)j0 (a) (b) (c) (d)

Figure 5.1: Possible boundary disturbances responsible for reconnection at a 3D positive null, showing the di- rection of the induced current density, for (a) torsional-spine reconnection: a rotation of the fan (red) induces a unidirectional current density, parallel (below the fan) and antiparallel (above the fan) to the spine, (b) torsional- fan reconnection: opposite rotations above and below the fan of the field lines about the spine (yellow) induce a current density antiparallel to the spine, (c) and (d) fan-spine reconnection: a shear of the fan plane (blue) or the spine (green), induces a current density perpendicular to the spine in these cases along thex-axis.

current density. This is called torsional fan reconnection. As before it does not produce a change in the topology of the field, and reconnection does not happen at the locations of the null (Pontin et al., 2004; Wyper and Jain, 2010).

3) A shear motion of the spine below and above the fan, in opposite directions (green arrows in Figure 5.1), or a shearing of the fan plane (blue arrows in Figure 5.1), producing a tilt with respect to the spine about a given axis, drives a collapse of the null point. That is, the resulting Lorentz forces act in the same direction as the initial disturbance, thus increasing it and resulting in a folding of the spine and fan towards each other. A current is created along the line to which the spine and fan are collapsing to, and so, it is perpendicular to the direction of the perturbation, similarly to the two-dimensional X-point collapse. Here, reconnection can take place in the vicinity of the null, and implies that flux is transferred across the spine and the fan, thus changing the global topology of the field lines, as in the two-dimensional case. Pontin et al. (2005) give a model for this type of reconnection, referred to as fan-spine reconnection.

All the previously mentioned studies of 3D reconnection at magnetic nulls assume a zero beta plasma model, solving the equations only for the electromagnetic field, and hence, neglecting the effects of the plasma in the evolution of the field. On the other hand, Pontin et al. (2007a) investigated current sheet formation and evolution of the field at 3D nulls after a shearing-type perturbation, using a full MHD description of the field. They then studied the subsequent reconnection processes using full MHD resistive numerical simulations, finding, at the time of maximum current, the biggest current accumulation at the location of the null, extended faintly along the spine

and the fan. Using the same MHD approach, they investigated the effects of compressibility (Pontin et al., 2007b) in their evolution, finding a significant reduction of the peak current and the reconnection rate as the limit of an incompressible fluid was approached.

However, the non-resistive evolution of three-dimensional nulls through a shearing-type perturbation and the development of singular currents, to our knowledge, has only been studied by Pontin and Craig (2005), who analysed the formation of a current singularity at the location of the null in a non-force-free equilibrium, in an equivalent manner to the two-dimensional singularities studied by Craig and Litvinenko (2005). The emphasis of their study was the evaluation of the scaling laws for the strength of the singularity, as a function of the grid resolution of their experiments. They also studied the effects of the plasma pressure in the relaxation, finding that, while a singularity was formed in all cases, the plasma pressure force weakened the strength of the singularity. For the evolution of the field, they assumed the adiabatic polytropic modelp∼ργ_{, using a frictional code where no}

energy conversion was allowed.

Here, we study the non-resistive evolution of two configurations, 1) a torsional-spine-type and 2) a sheared-type perturbed magnetic null, using LARE3D. In particular, we are interested in the current accumulations that arise when a non-force-free equilibrium is reached. We evaluate the effects of both the plasma pressure and the heat transfer in the evolution, as both the initial disturbance (i.e. the torsion or the shear) and the background plasma pressure are changed systematically. In the case of a shearing perturbation, the formation of a current singularity at the location of the null is evaluated.