# MHD reconnection

Post-publication activity

Curator: Eric R Priest

Figure 1: Breaking and reconnection of magnetic field lines when a localised diffusion region (shaded) leads to a change of connectivity of plasma elements (AB to AC)

Magnetic reconnection is a change of magnetic connectivity of plasma elements due to the presence of a localised diffusion region, where the magnetic field may diffuse through the plasma (Figure 1). Reconnection is a fundamental process in an almost-ideal plasma whose magnetic Reynolds number $$R_{me}=L_{e}v_{e}/\eta,$$ is much larger than unity, where $$L_{e}$$ is the typical global length-scale for variations in plasma properties, $$v_{e}$$ is a typical plasma velocity and $$\eta$$ is the magnetic diffusivity.

On Earth, one is surrounded by the three normal states of matter, namely, solids, liquids and gases, but, if the temperature of a gas is too high, it becomes a plasma or ionised gas, the fourth state of matter. Most of the universe is in this plasma state, and the main way that a plasma differs from a normal gas is in the subtle and complex interaction it has with a magnetic field, responsible for much dynamic behaviour.

On large scales ($$L_{e}$$) the plasma behaves like an ideal medium, with little or no resistive dissipation of any form, and hence no significant magnetic diffusion and reconnection. So the plasma elements preserve their magnetic connections. On the other hand, magnetic stresses can create thin localized regions of small thickness $$l$$ ($$\ll L_{e}$$) say, wherein the steepened field gradients create intense electric currents and non-ideal effects, e.g. resistive dissipation, Hall effect, etc. become important. Here magnetic field line reconnection can take place, since the magnetic connectivity of the plasma elements is not preserved in the ensuing evolution.

For many purposes, the behaviour of plasma and magnetic fields is described by magnetohydrodynamics (MHD for short) and in this article we shall focus on the MHD of magnetic reconnection. MHD is a macroscopic theory that is valid when the smallest length-scale, namely, the width of the diffusion region, is larger than the mean-free path for collisions. When this condition fails, collisionless plasma processes come into play. For such non-MHD aspects see Birn and Priest (2007).

The usual principal effects of magnetic reconnection are:

• (i) to convert some of the magnetic energy into heat by ohmic dissipation;
• (ii) to accelerate plasma by converting magnetic energy into bulk kinetic energy;
• (iii) to generate strong electric currents and electric fields, as well as shock waves

and current filamentation, all of which may accelerate fast particles;

• (iv) to change the global connections of the field lines and so affect the paths of

fast particles and heat, which are directed mainly along the magnetic field.

This brief review describes 2D reconnection, which is well understood, followed by 3D reconnection, which is very much a matter of current research. For further details see Priest and Forbes (2000), Birn and Priest (2007) or Priest and Parnell (2010).

## Contents

### Brief History

Sweet (1958) and Parker (1957) presented a 2D model for slow reconnection – too slow for solar flares. Then Petschek (1964) proposed the first model for fast reconnection. Later, Priest and Forbes (1986) discovered a wider family of Almost-Uniform solutions, including Petschek's mechanism and Biskamp (1986)'s numerical experiments as special cases.

It is now a well established part of reconnection theory that, when the magnetic diffusivity is enhanced at the reconnection point, Petschek's mechanism and the other Almost-Uniform reconnection regimes can indeed occur, and that an enhancement of diffusivity is a common effect in practice.

The emphasis is now focussed on 3D reconnection, which is completely different from 2D reconnection. In 3D several different regimes of reconnection have been proposed, including quasi-separatrix layer (QSL) reconnection in the absence of a null point, separator reconnection at a field line that joins two null points, and three kinds of reconnection at a null point, namely, torsional spine reconnection, torsional fan reconnection and spine-fan reconnection.

### MHD Equations

The induction equation $\tag{1} \frac{\partial \vec B }{ \partial t} = \nabla \times (\vec v \times \vec B) + \eta \nabla^{2} {\vec B},$

describes how the magnetic field $$(\vec B)$$ changes in time due to advection at velocity $$(\vec v)$$ of the field with the plasma and diffusion through the plasma where $$\eta$$ is the magnetic diffusivity (here assumed uniform). The ratio of these terms is the magnetic Reynolds number $$R_{m} = l_{0} V_{0} / \eta\ .$$

The other main equation for steady-state MHD reconnection (when the dominant forces are a pressure gradient and a magnetic force) is the equation of motion $\tag{2} \rho (\vec v \cdot \nabla) \vec v = - \nabla p + {\vec j \times \vec B},$

where $$\rho$$ and $$p$$ are the plasma density and pressure. These are supplemented by an energy equation and a mass continuity equation.

In MHD the electric current density ($$\vec j$$) and electric field ($$\vec E$$) are secondary and are given by $\vec j = \nabla \times \vec B / \mu\ \ \ \ \ {\rm and}\ \ \ \ \ \vec E + \vec v \times \vec B = \eta \nabla \times \vec B.$ Usually diffusion is negligible and the electric field equation reduces to $$\vec E + v \times \vec B = 0.$$

### Classes of Magnetic Field Evolution

New concepts arise in moving from 2D to 3D. When nonideal effects are important in a localised region, several classes of evolution of a magnetic field satisfy Faraday's law ($${\partial \vec B} / {\partial t} = -{ \nabla} \times \vec E$$) and $${ { \nabla}} \cdot \vec B=0$$ (Figure 2). The largest subclass conserves electromagnetic flux ($$\int_{S(t)} \vec B\! \cdot \! d{\vec S} + \int_{S(t)} \vec E\! \cdot \! d{\vec l}\ dt = constant$$). One subclass conserves magnetic flux ($$\int_{S(t)} \vec B\cdot d{\vec S} = const$$), while another represents 3D reconnection.

Figure 2: The different classes of evolution of a magnetic field. Within all possible types of evolution (indicated as "All"), there is a large class that conserves electromagnetic flux (called "E.m."). In turn, entirely within that class is a family of three-dimensionally reconnecting situations (called "3D rec") and a second family that conserves magnetic flux (called "Magnetic"). Two-dimensional reconnection (called "2D rec") is a special case of 3D reconnection that also conserves magnetic flux.

When $\frac{\partial \vec B }{ \partial t} = { { \nabla} } \times ({\vec v \times \vec B}) \ \ \ \ and \ \ \ \ \ \vec E + \vec v \times \vec B = {0},$ the magnetic flux and field line connections are both conserved and there is no reconnection, so that the magnetic topology is conserved. The term magnetic topology refers here to any property that is preserved by an ideal displacement, such as the linkage and knottedness of the field.

If instead the plasma is non-ideal with $\vec E + \vec v \times \vec B = {\vec N},$ where $$\vec N$$ represents any nonideal term such as $$\vec N = \eta \vec j\ ,$$ then the condition $$\vec B \times ({ { \nabla}} \times \vec N)=0$$ implies field-line conservation, whereas $${ { \nabla}} \times \vec N=0$$ implies flux conservation. Thus, line conservation and flux conservation are no longer equivalent. Although flux conservation implies field-line conservation, the reverse is not true.

## Two-Dimensional Null Points and Current Sheets

### 2D Null Points

Null points are locations where the magnetic field vanishes, and, in particular, X-points are potential weak spots, in the sense that current sheets tend to be created at them. In two dimensions reconnection can take place only at an X-point. The magnetic field near such a point may be written $\tag{3} B_{x} = B_{0} \frac{y}{y_{0}}, \qquad B_{y} = B_{0} \frac{x}{x_{0}},$

where $$B_{0}\ ,$$ $$x_{0}$$ and $$y_{0}$$ are all constant.

### Current Sheet Formation

If photospheric footpoints of a coronal field move, an X-point is likely to collapse to form a current sheet, which can then diffuse away and reconnect.

Figure 3: (a) Magnetic field near an X-type neutral point which evolves to a field with a current sheet having at its ends either (b) Y-points or (c) reversed currents and singularities

Consider an ideal planar motion of a planar field (3) with $$y_{0}=x_{0}\ ,$$ and suppose a current sheet forms with the field around the sheet containing no current. Then an elegant way of writing the new field ($$B_{x}, B_{y}$$) (Figure 3) is as a function of the complex variable $$z=x+iy\ ,$$ where the sheet is a cut in the complex plane. Initially, $\tag{4} B_{y} + iB_{x} = \frac{B_{0}}{x_{0}}z.$

Then, when a sheet stretches from $$z=-iL$$ to $$z=iL\ ,$$ the new field is $\tag{5} B_{y} + iB_{x} = \frac{B_{0}}{x_{0}}(z^{2} + L^{2}) ^{1/2}.$

Extensions of this technique have been to current sheets: with singular ends; with curvature; in three dimensions; in non-potential fields; at separatrices due to shearing out of the plane; and in braiding geometries.

### Magnetic Annihilation

Magnetic annihilation is an important ingredient of magnetic reconnection, referring to the carrying in and cancelling of oppositely directed straight field lines.

A current sheet naturally tends to diffuse outwards, and so a steady state may be set up if magnetic flux is carried in at the same rate as it is trying to diffuse (Figure 4). A simple 2D incompressible model has a stagnation-point flow (with $$V_{0}/a$$ constant) $\tag{6} v_{x} =-\frac{V_{0}\ x }{ a}, \quad v_{y} = \frac{V_{0} \ y }{ a}.$

Figure 4: (a) Stagnation-point flow creating a steady current sheet of width $$2l$$(shaded). (b) Magnetic field profile, with small-$$x$$ and large-$$x$$ approximations shown dashed

The effect of such a flow on a unidirectional magnetic field $$\vec B_{y} = B(x)$$ which reverses sign at $$x=0$$ is determined by Ohm's Law (2), namely, $\tag{7} E- \frac{V_{0}x }{ a} B= \eta \, \frac{dB }{ dx},$

which may be solved for $$B(x)\ .$$ This exact solution of MHD may be generalised to a 3D flow or to a Reconnective Annihilation model (Craig and Henton, 1995).

## Two-Dimensional Reconnection

### Slow Reconnection: the Sweet-Parker Mechanism

Sweet and Parker determined the speed ($$v_{i}$$) with which field lines are carried into a steady diffusion region of length 2L and width 2$$l\ ,$$ say, occupying the whole boundary between two opposing magnetic fields (Figure 5).

First of all, for a steady state, magnetic field of strength $$B_{i}$$ is carried in at the same speed ($$v_{i}$$) as it is trying to diffuse outward, so that $\tag{8} v_{i} = \frac{\eta }{ l}.$

Figure 5: Sweet-Parker reconnection. The diffusion region is shaded. The plasma velocity is indicated by solid-headed arrows and the magnetic field lines by light-headed arrows

Secondly, the rate $$(4 \rho Lv_{i})$$ at which mass is entering the sheet must equal the rate $$(4 \rho \, l v_{o})$$ at which it is leaving, so that, if the density is uniform, $\tag{9} L \, v_{i} = l \, v_{o},$

where $$v_{o}$$ is the outflow speed. The width $$(l)$$ may now be eliminated between Eqs.(8) and (9) to give the inflow speed as $$v_{i}^{2} = \eta\ v_{o}/L.$$ If the plasma is accelerated along the sheet by a Lorentz ($${\vec j \times \vec B}$$) force, the outflow speed ($$v_{o}$$) is the Alfven speed at the inflow, namely, $$v_{o} = v_{Ai} =B_{i}/\sqrt{\mu \rho}$$ and the reconnection rate is $\tag{10} v_{i} = \frac{v_{Ai} }{ R_{mi} \,^{1/2}}$

in terms of the magnetic Reynolds number $$(R_{mi} =Lv_{Ai}/\eta),$$ or in dimensionless form $M_{i}=R_{mi}^{-1/2},$ where $$M_{i}=v_{i}/v_{Ai}$$ is the inflow Alfven Mach number.

In the Sweet-Parker mechanism, we identify the sheet length $$(L)$$ with the global external length-scale $$(L_{e})$$ and $$R_{mi}$$ therefore with the global magnetic Reynolds number $$R_{me} = {L_{e} \, v_{Ae} / \eta}.$$ Since in practice $$R_{me} \gg 1,$$ the reconnection rate is very small: for instance, in the solar corona where $$R_{me}$$ lies between, say, $$10^{6}$$ and $$10^{12}\ ,$$ the fields reconnect at between $$10^{-3}$$ and $$10^{-6}$$ of the Alfven speed – much too slow for a flare.

Important extra effects are those of compressibility, which slows down the outflow when $$\rho_{o} > \rho_{i}\ ,$$ and a pressure gradient along the diffusion region, which can also slow the outflow when the outlfow pressure ($$p_{o}$$) is large enough.

### Fast Reconnection: Petschek's Model

Petschek suggested that the Sweet-Parker diffusion region is limited to a small segment (of length $$L\ll L_{e}$$) of the boundary between opposing fields. The diffusion region is thinner and so reconnection can take place faster.

The term fast reconnection refers to cases with a reconnection rate $$(M_{e} = v_{e}/v_{Ae})$$ much greater than the Sweet-Parker rate (10). There are different models for the external region, with Petschek's mechanism as one example.

Petschek (1964) realised that a slow-mode shock provides another way (as well as a diffusion region) of converting magnetic energy into heat and kinetic energy. He suggested that four such shocks would stand in the flow when a steady state is reached. Indeed, most of the energy conversion takes place at the shocks (Figure 6a).

Figure 6: Petschek's model, in which the central shaded region is the diffusion region and the other two shaded regions represent plasma that is heated and accelerated by the shocks

The terms "potential", "nonpotential", "uniform" and "nonuniform" are used to refer to the nature of the magnetic field in the inflow region upstream of the diffusion region and slow-mode shocks. Petschek's regime is almost-uniform in the sense that the field in the inflow region is a small perturbation to a uniform field $$(B_{e})\ .$$ It is also potential in the sense that there is no current in the inflow region.

In the Petschek analysis the magnetic field decreases from a uniform value $$(B_{e})$$ at large distances to a value $$B_{i}$$ at the entrance to the diffusion region given by $\tag{11} B_{i} = B_{e} \left( 1-\frac{4M_{e} }{ \pi} \log \frac{L_{e} }{ L} \right).$

Petschek suggested that the mechanism chokes itself off when $$B_{i} = \frac{1 }{ 2} B_{e}$$ in (11), which gives a maximum reconnection rate $$(M_{e}^{*})$$ of $\tag{12} M_{e}^{*} \approx \frac{\pi }{ 8 \log R_{me}}.$

This lies in practice between 0.1 and 0.01, and so is much faster than Sweet-Parker's model.

### Almost-Uniform Reconnection

Petschek's mechanism has been generalised by adopting different boundary conditions. Priest and Forbes (1986) sought fast, steady, almost-uniform reconnection solutions to the MHD equations for two-dimensional, ideal, incompressible flow and, by allowing pressure gradients in the inflow region (represented by a parameter $$b$$). They discovered a new family of regimes with a rich diversity of properties, depending on the nature of the flow on the inflow boundary. Petschek $$(b=0)$$ reconnection is just one particular member of a much wider class.

The almost-uniform theory has been compared with a variety of numerical experiments (Forbes, 1987). It has also been extended when the inflow magnetic field possesses highly curved field lines to give regimes of Nonuniform Reconnection (Priest and Lee, 1990). The numerical experiments confirm that fast reconnection does indeed exist, provided there are appropriate boundary conditions and a locally enhanced magnetic diffusivity in the diffusion region, due to, say, current-induced micro-instabilities, which are highly likely in solar coronal, space and astrophysical current sheets. Therefore, fast reconnection is a prime candidate for rapid energy conversion in solar, space and astrophysical plasmas.

One aspect of the simulations remains puzzling. When $$\eta$$ is uniform, the steady-state solutions are usually no longer sustained. Baty et al (2009a,b) have presented an example of fast reconnection with a uniform resistivity, but they suggested that usually such a state is marginally stable.

### Value of the Reconnection Rate

An essential feature of magnetic reconnection is the rate. We have seen above how the rate of reconnection has been calculated for the Sweet-Parker model, the Petschek model and the Almost-Uniform Family of models. The characteristic diffusion time across a scale $$l$$ is $\frac{l^{2}}{\eta},$ where $$\eta$$ is the magnetic diffusivity. For ionized hydrogen, Ohmic diffusion in the corona provides a value $$\eta=10^{9}T^{-3/2}\ m^{2}s^{-1}$$ for a temperature $$T\ .$$ This is a lower limit on the effective diffusion process, because of the possibility of plasma turbulence. In any case, it is apparent that the process of magnetic diffusion often proceeds too slowly in space and in the upper solar atmosphere. For instance, the coronal magnetic field is continually manipulated by the photospheric granule motions of 1 km/s on scales of 300 km and times scales of 300 s. If these motions produce scales of $$l = 10 km\ ,$$ say, they provide a characteristic diffusion time of 3 yrs. Comparing this with the characteristic dynamical time of 300 s for the granule motions suggests classical diffusion is too slow. By comparison, in the photosphere there is enough resistivity to reduce the characteristic diffusion time to much smaller values.

It is evident, then, that magnetic reconnection is of interest primarily in those situations where the magnetic stresses concentrate the magnetic field gradient into a very thin layer, providing much faster diffusion and reconnection than normal. The original Sweet-Parker reconnection scenario involved pressing together the bipolar fields of two sunspot pairs. The characteristic scale $$l$$ decreases without limit until the diffusion speed becomes as large as the inflow speed. The inflow speed is influenced by how fast the plasma can be expelled along the field lines to regions of lower pressure by the magnetic pressure. A steady state arises when the diffusion speed becomes equal to the inflow speed. The reconnection velocity, given by the inflow velocity across the broad scale $$L$$ of the magnetic field, has an Alfvenic Mach number of $$M_{i}=R_{mi}^{-1/2}\ .$$

Then the Almost-Uniform reconnection modes (including Petschek) confine the Sweet-Parker reconnection to a very small region near the neutral point and provide even faster reconnection, with an inflow Alfvenic Mach number of as much as 0.1. The conclusion is that the reconnection rate varies enormously, depending on the local circumstances. The theoretical work of Drake and coworkers (e.g., Drake et al, 1994, Phys. Rev. Letts. 73, 1251-1252) goes a step further to show that the magnetic stresses may squeeze the thickness of the current sheet to such small values as to approach the ion cyclotron radius and the ion inertial length, so that plasma kinetics determines the dissipation, with the ions decoupling from the magnetic field in the current sheet. Such treatment of the plasma kinetics shows greatly enhanced dissipation and reconnection rate, and is more favourable for the Almost-Uniform modes. That is to say, it supplies the enhanced dissipation around the neutral point necessary for fast reconnection to occur. These developments leave little doubt that rapid reconnection plays a role in most active dynamical situations.

The theory of dynamically driven reconnection in the diverse geometries of 3D magnetic fields is more complicated, as described below and is in a state of active exploration. It should be recognized that reconnection in 3D is in general very different from 2D. However, one possibility is to add a uniform magnetic field perpendicular to a 2D field, which just adds uniform pressure to the system, so that the Alfvenic Mach number of the inflow is unchanged. Another possibility is to flatten a separator and create locally a regime of Almost-Uniform reconnection, depending on local plasma conditions and upon the particular geometry and strength of the 3D magnetic field in which it arises.

### Tearing Mode Instability

Furth, Killeen, Rosenbluth (1963) discovered that a one-dimensional current sheet or a sheared magnetic field can go unstable to three kinds of resistive instability, one of which is called the tearing mode instability. They applied a wavelike perturbation to a static equilibrium and found conditions for an instability in which the magnetic fields reconnect in the centre of the system on a time-scale roughly of order $$\sqrt (\tau_d \tau_A)$$ in terms of the diffusion time ($$\tau_d$$) and the Alfven travel time ($$\tau_A$$). The outflow from a current sheet of finite width is found to have a stabilizing effect on the tearing instability (Bulanov, Sakai, Syrovatskii, 1979).

Numerical experiments have shown that the tearing mode can evolve nonlinearly into a regime of fast reconnection. Also, if the diffusion region in a state of fast steady reconnection (such as Sweet-Parker or Petschek or Almost-Uniform reconnection) becomes too long, it goes unstable to tearing and coalescence, to give what is called an impulsive bursty regime of reconnection (Priest, 1986b), characterized by a more rapid energy release in a series of bursts as the islands form and coalesce. The criteria for the onset of impulsive bursty reconnection have been discussed by Forbes & Priest (1987).

## The Geometry and Topology of 3D Magnetic Fields

### 3D Null Points

At a null point the magnetic field vanishes, and nearby a linear null point it increases linearly with distance. The simplest example is $(B_{x}, B_{y}, B_{z}) = (x,y, -2z),$ so that $${ \nabla \cdot \vec B} = 0$$ is satisfied identically.

Figure 7: (a) The structure of a 3D null point showing the spine and fan,
Figure 8: (b) a proper radial null with radial fieldlines in the fan and (c) an improper radial null with curved fieldlines in the fan

Two families of field lines, the spine and the fan, pass through a linear null (Priest and Titov, 1996). The spine is the isolated field line (Figure 7) lying along the $$z$$-axis. Its neighbouring field lines form two bundles which spread out as they approach the fan surface (the $$xy$$-plane). More general linear nulls (Parnell et al, 1996) include an oblique null and a spiral null.

### Separatrices, Separators and Skeletons

In three dimensions, complex configurations have separatrix flux surfaces or separatrices separating the volume into topologically different regions, in the sense that all the field lines in one region start at a particular source on the photosphere and end at a particular sink. They intersect each other in a separator, a special field line which ends at null points or on the boundary. Many separatrix surfaces are extensions of fan surfaces from null points, and others touch a boundary in a bald patch (Titov et al, 1993). Often, 3D reconnection involves the transfer of flux across separatrices from one region to another.

The atmosphere of the Sun is threaded by flux from myriads of sources where flux pokes through the photosphere into the overlying corona. A powerful way to understand the structure of such complicated fields is to construct the topological skeleton of the field (i.e., the web of separatrix surfaces) and also the quasi-skeleton of quasi-separatrix layers (QSLs). A QSL is a region in a magnetic field where the gradient of the footpoint mapping is large and they intersect in quasi-separators. The term structural skeleton (Titov et al, 2007, 2009) refers to the sum of both the separatrices and QSLs. Reconnection tends to occur wherever strong currents concentrate, especially near null points, and along separators and quasi-separators.

## Three-Dimensional Reconnection Concepts

There are subtle differences between diffusion and reconnection and between 2D and 3D reconnection. Consider a non-ideal plasma with $\tag{13} \vec E + \vec v \times \vec B = {\vec N}.$

The presence or absence of diffusion or reconnection depends on the nature of $$\vec N\ .$$ If it can be written as $$\vec N = {\vec u} \times \vec B + { \nabla} \Phi,$$ then the curl of (13) becomes $$\partial \vec B/\partial t = { { \nabla}} \times ({\vec w} \times \vec B),$$ where $${\vec w}= \vec v - \vec u$$ is a flux velocity with which the magnetic field moves and $$\vec u$$ is the slippage velocity. It follows that:

• (a) if $$\vec N = \vec u \times \vec B + { \nabla} \Phi$$ and $$\vec u$$ is smooth, then magnetic field slips with no reconnection;
• (b) if $$\vec N = \vec u \times \vec B + { \nabla} \Phi$$ and $$\vec u$$ is singular, then there is 2D reconnection;
• (c) if $${\vec N} \neq \vec u\times \vec B + { \nabla} \Phi\ ,$$ then there is reconnection in 2.5D or 3D.

Thus, reconnection involves diffusion, but diffusion can occur without reconnection.

### Reconnection in Two Dimensions $$(\vec E \cdot \vec B =0)$$

For 2D MHD, $$\vec E \cdot \vec B =0$$ and we may find a flux-preserving flow $$\vec w$$ such that $$\vec E + \vec w \times \vec B=0.$$ In this case, three types of behaviour are possible:

• (a) If $$\vec B \neq 0\ ,$$ then $$\vec w$$ is smooth everywhere and we have slippage of the magnetic field;
• (b) If $$\vec B = 0$$ at some point and the neighbouring magnetic field is elliptic, then magnetic flux is destroyed or generated at the null;
• (c) If $$\vec B = 0$$ at some point and the neighbouring magnetic field is hyperbolic, then there is magnetic flux reconnection, with the flux conserved.

However, in 3D for an isolated 3D nonideal region, a flux conservation velocity ($$\vec w$$) does not in general exist (Priest et al, 2003).

### Differences Between 2D and 3D Reconnection

Several important differences have emerged between 2D and 3D reconnection. In two dimensions: reconnection can occur only at an X-point; a flux velocity ($$\vec w$$) always exists and is singular at the X-point; and in the diffusion region, field lines slip through the plasma and change their connections only at the X-point.

In three dimensions, none of the above properties hold, so that: reconnection can occur at a null or in the absence of a null; a single flux velocity does not exist, but can be replaced by a dual flux velocity, namely, a pair ($$\vec w_{in},\vec w_{out}$$) of flux velocities that describe separately the parts of a field line that enter and leave a diffusion region; in the diffusion region, field lines continually change their connections.

### Definition and Classification of Reconnection

Schindler et al (1988) suggested a concept of "General Magnetic Reconnection", which includes all effects of local nonidealness that produce a component $$(E_{||})$$ of the electric field along a particular magnetic field line, so that $\tag{14} \int E_{||} \, ds \neq 0$

is a necessary and sufficient condition for such reconnection, where the integral is taken along that field line. A natural way of classifying non-ideal processes is given in Figure 9, partly using Schindler's ideas.

Figure 9: Classification of non-ideal processes

Thus, 3D reconnection is defined as happening when there is a change of magnetic connectivity of plasma elements and is diagnosed by the condition (14). The precondition for such reconnection is the formation of a localised current concentration. Indeed, the key reason why nulls, separators and QSLs are natural locations for reconnection is that they are locations where strong currents tend to grow.

## Three-Dimensional Reconnection Regimes

### Three-Dimensional Null Regimes

Reconnection can occur at a null point, a separator or a non-null region. From the kinematics of steady ideal flows (satisfying $$\vec E+\vec v \times \vec B=0 \ {\rm and} \ { { \nabla}} \times \vec E = 0$$) in the neighbourhood of a null point or separator, three types were proposed, depending on where the current concentrates, namely spine reconnection, fan reconnection and separator reconnection (Priest and Titov, 1996).

Later, the nature of the flow in the diffusion region was studied (Hornig and Priest, 2003) and computational experiments Galsgaard et al, 2003, Pontin and Galsgaard, 2007a) led to a new categorisation into torsional spine reconnection, torsional fan reconnection and spine-fan reconnection (Priest and Pontin, 2009).

Figure 10: Rotational motions (open arrows) of: the fan driving torsional spine reconnection with a strong spine current (solid arrows)
Figure 11: Rotational motions of the spine driving torsional fan reconnection with a strong fan current and slippage of field lines (solid arrow)

In torsional spine reconnection, rotating the fan plane of a spiral null drives a current along the spine and creates twisting flows about it. Inside the spine current tube, there is rotational slippage, with the field lines becoming disconnected and rotating around the spine (Figure 10).

Torsional fan reconnection is also caused by twisting motions, namely, a rotation of the spine in opposite directions above and below the fan. This builds up a fan current sheet, in which field lines experience rotational slippage above and below the fan (Figure 11), but there is no flow across spine or fan.

The generic reconnection mode observed in numerical experiments in response to shearing motions is, however, spine-fan reconnection, which possesses a strong fan-aligned current with flow across both spine and fan, and is in some sense a combination of spine and fan reconnection (Pontin et al, 2005). It may be modelled with a null point having a fan-aligned current ($$B_{0}{\bar j_{0}}/(\mu L_{0})$$) in the $$x$$-direction and field components $$(B_{x},B_{y},B_{z})=(B_{0}/L_{0})(x,y-{\bar j_{0}}z,-2z).$$ Field lines traced from footpoints anchored in the fan-crossing flow are found to flip up and down the spine, whereas those that are traced from the top and bottom of the domain flip around the spine in the fan plane. It is the flux transfer across both spine and fan that distinguishes spine-fan reconnection, together with the presence of a current sheet that is inclined at an intermediate angle between the spine and fan.

Figure 12: The structure of spine-fan reconnection, showing field lines and (shaded) diffusion region
Figure 13: The corresponding motion of flux across spine and fan (large light arrows). The sheet current flows in the $$x$$-direction (large dark arrows)

### Separator and QSL Reconnection

The fans of two nearby null points will in general intersect in a special curve called a separator, which is a field line that joins one null to the other (Figure 14) and is a natural location for current sheet formation.

Figure 14: Configuration with two nulls including a separator joining one null to the other

In their numerical experiment on the response of a set of null points to footpoint shearing, Galsgaard and Nordlund (1997) found reconnection in current sheets forming along separators. Then Parnell and Galsgaard (2004), Parnell et al (2008a) modelled coronal reconnection driven by the motion of a pair of opposite-polarity photospheric fragments in an overlying horizontal field. They found the main dissipation mechanism to be separator reconnection at multiple separator current sheets, with dissipation enhanced by flux recycling in recursive reconnection. Furthermore, when the separator current is strong enough, the field in transverse planes changes from X-type to O-type. Dissipation by separator reconnection has been has been applied to coronal heating (Longcope, 2001, Priest et al, 2005) and solar flares (Longcope and Beveridge, 2007).

At quasi-separators reconnection of a similar type to separator reconnection may take place (Demoulin et al, 1996, Titov, 2007, Titov et al, 2009). QSL reconnection has been applied to models of 3D twisted flux tubes in solar flares (Demoulin et al 1996, 1997) and has been also been called slip-running reconnection by Aulanier et al (2006), Pariat et al (2006).

## References

• Aulanier, Guillaume; Pariat, Etienne; Demoulin, Pascal and Devore, Rick (2006). Slip-running reconnection in quasi-separatrix layers. Solar Phys. 238: 347–376. doi:10.1007/s11207-006-0230-2.
• Baty, Hubert; Priest, Eric and Forbes, Terry (2009). Effect of nonuniform resistivity in Petschek reconnection. Phys. Plasmas 13: 022312/1–7. doi:10.1063/1.2172543.
• Baty, Hubert; Priest, Eric and Forbes, Terry (2009). Petschek reconnection with a nonlocalized resistivity. 'Phys. Plasmas 16: 012102/1–6. doi:10.1063/1.3062833.
• Biskamp, Dieter (1986). Magnetic reconnection via current sheets. Phys. Fluids 29: 1520. doi:10.1063/1.865670.
• Bulanov, S; Sakai, DI and Syrovatsky, SI (1979). Tearing instability in quasi-stationary magnetohydrodynamic configurations. Soviet Journal of Plasma Physics 5: 280–290.
• Craig, Ian and Henton, S (1995). Exact solutions for steady state incompressible magnetic reconnection. Astrophys. J. 450: 280–288. doi:10.1086/176139.
• Demoulin, Pascal; Bagala, L; Mandrini, Christina; Henoux, Jean-Claude and Rovira, Martira (1997). Quasi-separatrix layers in solar flares. II. Observed magnetic configurations. Astron. Astrophys. 325: 305–317.
• Demoulin, Pascal; Henoux, Jean-Claude; Priest, Eric and Mandrini, Christina (1996). Quasi-separatrix layers in solar flares. I. Method. Astron. Astrophys. 308: 643–655.
• Forbes, Terry and Priest, Eric (1987). A comparison of analytical and numerical models for steadily-driven magnetic reconnection. 'Rev. Geophys. 25: 1583–1607. doi:10.1029/rg025i008p01583.
• Furth, HP; Killeen, J and Rosenbluth, MN (1963). Finite-resistivity instabilities of a sheet pinch. Phys. Fluids 6: 459–484. doi:10.1063/1.1706761.
• Galsgaard, Klaus and Nordlund, Aake (1996). Heating and activity of the solar corona. III. Dynamics of a low-beta plasma with 3D null points. J. Geophys. Res. 102: 231–248. doi:10.1029/96ja02680.
• Galsgaard, Klaus; Priest, Eric and Titov, Slava (2003). Numerical experiments on wave propagation towards a 3D null point due to rotational motions. J. Geophys. Res. A1: SSH 10.1–10.9. doi:10.1029/2002ja009393.
• Hornig, Gunnar and Priest, Eric (2003). Evolution of magnetic flux in an isolated reconnection process. Phys. Plasmas 10: 2712–2721. doi:10.1063/1.1580120.
• Longcope, Dana (2001). Separator current sheets: Generic features in minimum-energy magnetic fields subject to flux constraints. Phys. Plasmas 8: 5277–5289. doi:10.1063/1.1418431.
• Longcope, Dana and Beveridge, Colin (2007). A quantitative, topological model of reconnection and flux rope formation in a two-ribbon flare. Astrophys. J. 669: 621–635. doi:10.1086/521521.
• Pariat, Etienne; Aulanier, Guillaume and Demoulin, Pascal (2006). A new concept for magnetic reconnection: slip-running reconnection. Semaine de l'Astrophysique Francaise, Paris. 559-562
• Parker, Gene (1957). Sweet's mechanism for merging magnetic fields in conducting fluids. J. Geophys. Res. 62: 509-520. doi:10.1029/jz062i004p00509.
• Parnell, Clare and Galsgaard, Klaus (2004). Elementary heating events - interaction between two flux sources. II Rates of flux reconnection. Astron. Astrophys. 428: 595–612. doi:10.1051/0004-6361:20034350.
• Parnell, Clare; Haynes, Andrew and Galsgaard, Klaus (2008). Recursive reconnection and magnetic skeletons. Astrophys. J. 675: 1656–1667. doi:10.1086/527532.
• Parnell, Clare; Smith, Jason; Neukirch, Thomas and Priest, Eric (1996). The structure of 3D magnetic neutral points. Phys. Plasmas 3: 759–770. doi:10.1063/1.871810.
• Petschek, Harry (1964). Magnetic field annihilation. AAS-NASA Symposium on the Physics of Solar Flares, Washington. 425–439
• Pontin, David and Galsgaard, Klaus (2007). Current amplification and magnetic reconnection at a three-dimensional null point: Physical characteristics. J. Geophys. Res. 112: A03103. doi:10.1029/2006ja011848.
• Pontin, David; Hornig, Gunnar and Priest, Eric (2005). Kinematic reconnection at a magnetic null point: Fan-aligned current. Geophys. Astrophys. Fluid Dyn. 99: 77–93. doi:10.1080/03091920512331328071.
• Priest, Eric and Forbes, Terry (1986). New models for fast, steady-state reconnection. J. Geophys. Res. 9: 5579–5588. doi:10.1029/ja091ia05p05579.
• Priest, Eric (1986). Magnetic reconnection on the Sun. Mit. Astron. Ges. 65: 41–51.
• Priest, Eric; Hornig, Gunnar and Pontin, David (2003). On the nature of three-dimensional magnetic reconnection. J. Geophys. Res. A7: SSH 6.1–6.8. doi:10.1029/2002ja009812.
• Priest, Eric and Lee, Lou (1990). Nonlinear magnetic reconnection models with separatrix jets. J. Plasma Phys. 44: 337–360. doi:10.1017/s0022377800015221.
• Priest, Eric; Longcope, Dana and Heyvaerts, Jean (2005). Coronal heating at separators and separatrices. Astron. Astrophys. 624: 1057–1071. doi:10.1086/429312.
• Priest, Eric and Pontin, David (2009). Three-dimensional null point regimes. Phys. Plasmas 16: 122101. doi:10.1063/1.3257901.
• Priest, Eric and Titov, Slava (1996). Magnetic reconnection at three-dimensional null points. Phil. Trans. Roy. Soc. Lond. 355: 2951–2992. doi:10.1098/rsta.1996.0136.
• Schindler, Karl; Hesse, Michael and Birn, Joachim (5547–5557). General magnetic reconnection, parallel electric fields, and helicity. 93 1988: J. Geophys. Res..
• Electromagnetic Phenomena in Cosmical Physics, Sweet (1958). The neutral point theory of solar flares. Cambridge University Press, Cambridge. 124–134
• Titov, Slava (2007). Generalized squashing factors for covariant description of magnetic connectivity in the solar corona. Astron. Astrophys. 660: 863–873. doi:10.1086/512671.
• Titov, Slava; Forbes, Terry; Priest, Eric; Mikic, Zoran and Linker, Jon (2009). Slip-squashing factors as a measure of three-dimensional magnetic reconnection Astron. Astrophys. 693: 1029–1044. doi:10.1088/0004-637x/693/1/1029.
• Titov, Slava; Priest, Eric and Demoulin, Pascal (1993). Conditions for the appearance of bald patches at the solar surface. Astron. Astrophys. 276: 564–570.

• Birn, Joachim and Priest, Eric (2007). Reconnection of Magnetic Fields: MHD and Collisionless Theory and Observations. Cambridge University Press, Cambridge.
• Priest, Eric and Forbes, Terry (2000). Magnetic Reconnection: MHD Theory and Applications. Cambridge University Press, Cambridge.
• Priest, Eric and Parnell, Clare (2011). MHD Reconnection Theory Living Reviews in Solar Physics, Lindau.