1.5 Current sheets and reconnection
1.5.2 Current sheet formation
The analytical form of current sheets in two-dimensional fields, created following the collapse of a hyperbolic X-point, were firstly studied by Green (1965), who suggested an expression for a one-dimensional current sheet of the form
By+iBx=
p
Z2+a2, (1.5.1)
whereZ =x+iyrepresents the complex plane, and2ais the length of the sheet. The four separatrices open from both ends of the current sheet in two so called Y-points, inclined to one another with an angle of2π/3(see Figure 1.12 for a schematic categorization of the possible magnetic 2D singular points described in this section).
(a) X-point (b) Y-point (c) cusp-point (d) T-point
Figure 1.12: Special magnetic points in two-dimensions.
Somov and Syrovatskii (1976) described the collapse of a two-dimensional X-point with a more general solu- tion given by
By+iBx= Z
2+l2
√
Z2+a2 , (1.5.2)
wherel2 < a2. The two null points at the ends of the sheets are singular. This case reduces to Green’s solution whenl2=a2.
Later on, Bungey and Priest (1995) extended the solution of Somov & Syrovatsky, providing an analytical expression for all the possible potential and force-free configurations around a linear current sheet,
By+iBx=−B0 bd2+ 2dcZ−Z2+1 2d2 √ Z2+a2 , (1.5.3)
whereb,c,dandB0are constants.
In all these cases, the current sheet is assumed to be infinitesimally thin, and the current density has aδ-like singularity across the sheet.
The above theory has been applied to more general planar current sheets in the potential and force-free solar corona, involving the magnetic field associated with two bipolar regions, by Priest and Raadu (1975) and Tur and Priest (1976), as in Figure 1.13. In these models, a curved current sheet replaces the linear sheet found in previous studies of X-point collapse, and the extremes of the sheet show a pair of cusp points (see Figure 1.12c), where the separatrices are curved in space. This configuration has been used in a variety of models of equilibria for solar coronal magnetic arcades and loops by Low (1981, 1982b, 1986). Also, Vainshtein (1990) and Vekstein and Priest (1993) tried to give analytical expressions for magnetic fields near special points, such as cusp points, assuming a potential, and force-free solution outside and inside the cusp, respectively, in the first case, and a MHS combined with a potential solution in the second case.
Before that, Parker (1972) considered the evolution of three dimensional braided magnetic flux tubes, finding rapid dissipation and reconnection, which enabled the topology of the magnetic field to reduce to a simple equi- librium form. Parker suggested that to first order, changes in pressure along a flux tube would only modify the verticalz-component of the field, and in general, “the pattern of the field does not vary along the general direction of the field”, in other words, an equilibrium exists only if the variations in the field consist of simple twisting of the
x y
-a +a
Figure 1.13: Schematic representation of the magnetic field configuration in the plane perpendicular to two line dipoles atx=±a, based on Tur and Priest (1976), Fig. 1, after an increase in the moment of the smaller dipole, creating a current sheet, here shown as a thick curve.
lines. In a more complex topology case, such as braided flux tubes wrapped around each other, he suggested that no equilibrium field was possible, and current would form at the boundaries of the tubes, leading to topological dissipation and merging of field lines in the process know as magnetic reconnection. Syrovatskii (1978) also sug- gested that the problem of continuous deformation of such fields had no solution in general. However, these results were disputed by van Ballegooijen (1985), who argued against Parker’s scheme. Instead he suggested that an equi- librium should always exist, without the need for any form of symmetry of the field, implying that the coronal field adjusts itself to the motions of the photosphere, and that current sheets are a result of photospheric motions and would appear only when the boundaries have discontinuities. More recently, the properties of three-dimensional current sheets have been developed by Longcope (1996, 1998).
All the above studies involve potential and force-free solutions and, in fact, the thin current sheet configurations from Bungey and Priest (1995) are not in equilibrium, even if the regions around them are. This is because the current varies along the sheet, but there is no plasma pressure to hold the Lorentz force within the current sheet. More recently, Rast¨atter et al. (1994), Craig and Litvinenko (2005) and Pontin and Craig (2005) have studied the magnetohydrostatic relaxation of X-type null points, considering plasma pressure forces, reaching a cusp-like equilibrium sheet with the Lorentz force being balanced by the plasma pressure gradient. Common features of these studies are the appearance of current accumulations along the field separatrices. Also, they find evidence to suggest that a singularity in current is formed at the location of the null, as in the potential and force-free cases, whose nature is unknown. Craig and Litvinenko (2005) find the plasma pressure to be enhanced in the regions inside the cusps, and decreased in the regions outside the cusps, as sketched in Priest and Forbes (2000). Figure 1.14 show some schematic views of the different two-dimensional sheet configurations coming from the collapse of a magnetic X-point.
(a) Potential X-point (b) Green´s current sheet
(d) Pair of cusp points (c) Cusp point
Figure 1.14: Sketch of two-dimensional equilibria, based on Priest and Forbes (2000), Fig. 2.10. Thick curves represent current sheets, and shaded regions are regions of enhanced plasma pressure. In the absence of plasma, if a hyperbolic X-point (a) is squashed in the vertical direction, a current sheet is formed, with Y-points at both ends, as in (b). In the presence of a non-zero beta plasma, if the X-point in (a) is squashed, the pressure is enhanced in the shaded regions, producing the equilibrium in (c), where a pair of cusp-points have formed, preserving the X-point geometry at the center, while, in a non-zero beta plasma, (b) produces the equilibrium in (d), where the current sheet has developed two cusp points at its ends, in which the pressure is enhanced.