1 Introduction

Magnetic reconnection is a fundamental process responsible for many dynamic phenomena in solar physics. It occurs in any plasma that is almost-ideal, in the sense that the global Lundquist number (i.e., the magnetic Reynolds number based on the Alfvén speed)

$$\begin{aligned} R_{me}\equiv S\equiv \frac{L_e v_{Ae}}{\eta } \end{aligned}$$

is much greater than unity, where \(L_{e}\) is the global length scale, \(B_{e}\) is the corresponding magnetic field, \(\eta \) is the magnetic diffusivity, and \(v_{Ae}=B_e/\sqrt{\mu \rho }\) is the global Alfvén speed. In this article we give a review of the MHD aspects of reconnection theory and refer the reader for further details to Priest and Forbes (2000), Birn and Priest (2007), Yamada et al. (2010) and Priest (2014), including collisionless theory and observational effects of reconnection. Many aspects that have been fully developed in Priest (2014) are treated more briefly here, but newer aspects are added including the magnetic topology of global coronal magnetic fields (Sects. 2.5, 2.7), current sheet formation in 3D magnetic fields (Sects. 5.35.6), the plasmoid instability (Sect. 8.3), fast reconnection in a collisional or collisionless medium (Sect. 9), new aspects and models of 3D reconnection and its implications for coronal dynamics (Sects. 1014), reconnection during flux emergence (Sect. 15.1.2), interchange reconnection (Sect. 15.2), and brief overviews of our understanding of 3D reconnection in the Earth’s magnetosphere (Sect. 15.3), in turbulence (Sect. 16) and non-thermal particle acceleration during reconnection (Sect. 17).

When the plasma is ideal (\(R_{me}\rightarrow \infty \)), the magnetic connections between plasma elements are preserved. However, when non-ideal effects in the induction equation come into play in a localised region of size L (\(\ll L_{e}\)), say, magnetic reconnection can occur—i.e., there can be a change of connectivity of plasma elements (as indicated in Fig. 1). Often the following physical effects are produced by magnetic reconnection:

  1. (a)

    the generation of strong electric currents, electric fields and shock waves, which in the solar atmosphere may accelerate fast particles;

  2. (b)

    ohmic dissipation of the currents, which transforms some of the magnetic energy into heat;

  3. (c)

    the appearance of strong Lorenz forces, which accelerate plasma to high speeds;

  4. (d)

    changes in the global connections of magnetic field lines, which alter the paths of fast particles and heat flow.

Resistive MHD provides a good model when the plasma is highly collisional, such as in the Sun’s interior and low atmosphere. Even in the collisionless outer corona, it still gives a reasonable model under certain caveats (see Birn and Priest 2007; Priest 2014), but Hall MHD with a two-fluid approach or a kinetic model provide fuller treatments, especially in the interiors of the tiny diffusion region and shock waves (Sect. 9).

Fig. 1
figure 1

For a related movie (courtesy of K. Galsgaard) see Supplementary Information

A change of magnetic connectivity is produced by reconnection in a localised diffusion region (shaded), such that a plasma element A is initially connected to a plasma element B but after reconnection it has become connected to C.

In this review, we first develop the background and fundamental concepts that are necessary for understanding the nature of reconnection (Sects. 15). We discuss the structure of null points, where the magnetic field vanishes, both in two dimensions (2D) (Sect. 2.1) and three dimensions (3D) (Sect. 2.2), as well as the ways in which such nulls collapse. Then we describe other geometrical features such as separatrices and quasi-separatrices, which map out the skeleton and quasi-skeleton of a complex magnetic configuration (Sects. 2.5,2.6). Other useful and subtle concepts are magnetic helicity (Sect. 4.6), the conservation of magnetic flux and field lines (Sect. 3.2), magnetic diffusion and field-line motion (Sect. 3.3). These enable us to go on to describe the different models for 2D and 3D reconnection in Sects. 612, as well as some applications and extensions in Sects. 1317.

1.1 Historical overview

Reconnection theory originated with: Giovanelli (1947)’s idea that electric fields near a magnetic neutral point may accelerate particles and generate heat in solar flares; Cowling (1953)’s realisation that a current sheet only a few metres thick and created by the collapse of an X-type neutral point could do so; and Dungey (1961)’s proposal of reconnection at the Earth’s Magnetosphere.

There have been four phases in the development of the theory, as follows.

  1. (i)

    The Sweet–Parker model (Sweet 1958a; Parker 1957) for steady-state reconnection in a thin current sheet of length L, in which magnetic field \(B_i\) is carried into the sheet at a speed

    $$\begin{aligned} v_{i}=\frac{v_{Ai}}{R_{mi}^{1/2}}, \ \ \ \ \ {\mathrm{where}} \ \ \ \ \ v_{Ai}=\frac{B_i}{\sqrt{\mu \rho }} \quad {\mathrm{and}} \quad R_{mi}=\frac{L v_{Ai}}{\eta } \end{aligned}$$

    are the inflow Alfvén speed and magnetic Reynolds number, respectively, based on \(B_i\), L and \(v_{Ai}\). When \(R_{mi}\gg 1\), the reconnection rate (\(v_{i}\)) is much smaller than \(v_{Ai}\), and so this model describes slow reconnection.

    Then Furth et al. (1963) discovered several resistive reconnection instabilities of a current sheet, including the tearing mode (Sect. 8). Also, Petschek (1964) proposed the first regime of fast reconnection, whose maximum reconnection rate is typically a hundredth or a tenth of the global Alfvén speed (\(0.1 v_{Ae}\)), and so it is indeed rapid enough for a solar flare. Most of the energy conversion takes place at four slow-mode shock waves that stand in the flow and extend outwards from a tiny central Sweet–Parker current sheet.

  2. (ii)

    Numerical experiments (Biskamp 1986) revealed solutions quite different from Petschek’s. These at first cast doubt on the validity of the Petschek mechanism, until Priest and Forbes (1986) discovered a whole family of Almost-Uniform models for fast reconnection, which include the solutions of both Petschek and Biskamp as special cases. The stability of these models has been clarified by Baty et al. (2009a, b), who found them to be stable when the magnetic diffusivity in the diffusion region is enhanced, which may well be produced by current-induced micro-turbulence. Such fast reconnection is thus one way to produce fast reconnection in the solar atmosphere.

  3. (iii)

    The realisation that fast reconnection at a similar rate to Petschek’s mechanism also occurs in two other situations, namely, collisionless reconnection and impulsive bursty reconnection. When reconnection is collisionless, the Hall effect creates an ion diffusion region of width equal to an ion inertial length together with a smaller electron diffusion region, and these replace the resistive diffusion region (Shay et al. 1998; Huba 2003). Here, the GEM Challenge has revealed that the same fast rate of reconnection is produced by full-particle, hybrid and Hall MHD codes (Birn et al. 2001). In impulsive bursty reconnection, the diffusion region becomes unstable to secondary tearing (Bulanov et al. 1979; Biskamp 1986; Priest 1986; Forbes and Priest 1987; Loureiro et al. 2007; Bhattacharjee et al. 2009).

  4. (iv)

    Most of the attention is now focused on 3D reconnection, which is revealing many new features that are completely different from 2D reconnection (Priest et al. 2003). A key realisation (Schindler et al. 1988) is that the condition for reconnection in 3D is the presence of an electric field (\(E_{\parallel }\)) parallel to the magnetic field, namely,

    $$\begin{aligned} \int E_{\parallel } \, ds \ne 0, \end{aligned}$$

    where the integral is taken along a magnetic field line through a diffusion region where the plasma is not ideal. If it is evaluated over all field lines, its maximum value determines the reconnection rate. Unlike the situation in 2D, this can occur in the absence of null points.

Various types of 3D reconnection have been discovered whenever strong localised currents form, depending on whether the current is concentrated along the spine or fan of a null point or along a separator (joining two null points) or a quasi-separator field line (i.e., a hyperbolic flux tube) or in a braid, namely:

  • torsional spine or torsional fan reconnection (Sect. 10.2) with rotational motions near a null;

  • spine-fan reconnection with shearing motions near a null (Sect. 10.2);

  • separator reconnection (Sect. 11) at the intersection of two separatrix surfaces;

  • quasi-separator or HFT reconnection (Sect. 12) at the intersection of two quasi-separatrix layers (QSLs);

  • and braid reconnection (Sect. 14.1).

Across a separatrix surface the mapping of magnetic field lines changes discontinuously, whereas across a QSL it changes extremely rapidly but continuously. Quasi-separator reconnection has also been referred to as slip-running reconnection (Aulanier et al. 2006), which refers to the magnetic flipping process (Priest and Forbes 1992) that is a common feature of much reconnection in three dimensions.

1.2 Summary of reconnection concepts

The behaviour of magnetic fields in 3D is much more subtle and complex than in 2D and exhibits many new features [see Sect. 4 and Priest (2014) for details]. If the plasma behaves in a nonideal way in a finite localised region, then 2D reconnection is only one of several classes of behaviour that obey Faraday’s law and \({{{\varvec{\nabla }}}} \cdot {\mathbf {B}}=0\). The largest subclass is the one that conserves electromagnetic flux (\(\int _{S(t)} {\mathbf{B}} \cdot d{\mathbf{S}} + \int _{S(t)} {\mathbf{E}} \cdot d{\mathbf{l}}\ dt = {\mathrm {const}}).\) Within that, there lie two large classes of solution, namely, those that conserve magnetic flux (\(\int _{S(t)} {\mathbf{B}}\cdot d{\mathbf{S}} = {\mathrm {const}}\)) and those that represent 3D reconnection Hornig, 2001. These in turn intersect in the subclass of 2D reconnection.

Magnetic flux is conserved when the magnetic flux through any surface moving with the plasma is constant, whereas magnetic field lines are conserved when any pair of plasma elements lying initially on a magnetic field line remain connected by a field line. For an ideal plasma, \({\mathbf{E}} + {\mathbf{v}} \times {\mathbf{B}} = {\mathbf{0}}\) and both of these conservation laws hold. A consequence of this is that the magnetic topology is also conserved, where magnetic topology refers to any property that is preserved during a smooth deformation, such as the linkage or knottedness of magnetic field lines.

However, when the plasma is non-ideal, so that \( {\mathbf{E}} + {\mathbf{v}} \times {\mathbf{B}} = {{\mathbf{N}}}, \) say, where \({\mathbf {N}}\ne 0\) represents any nonideal process, the physical effects depend on the form of \({\mathbf {N}}\). Thus, when \( {\mathbf {B}}\times ({{{\varvec{\nabla }}}} \times {\mathbf{N}})={\mathbf{0}}\) magnetic field lines are conserved, but when \( {{{\varvec{\nabla }}}} \times {\mathbf{N}}={\mathbf{0}}\) magnetic flux is conserved. The forms of these conditions imply that flux conservation and field-line conservation are no longer equivalent, in the sense that flux conservation implies field-line conservation, but the opposite is not true (Sect. 3.2).

There is an important distinction between magnetic diffusion and magnetic reconnection (Sect. 4.1). Reconnection is a global process which includes diffusion in a localised region, but there are examples of diffusion with no reconnection; for example, the magnetic field may diffuse though the plasma with plasma elements not changing their magnetic connectivity.

The form of \({\mathbf {N}}\) determines whether diffusion or reconnection occurs and also the type of reconnection. If it cannot be written in the general form \({\mathbf{N}} = {\mathbf{u}} \times {\mathbf{B}} + {\varvec{\nabla }}{\varPhi }\), then 2.5D or 3D reconnection takes place. However, if it can be written in this form, then:

(a) if \({\mathbf{u}}\) is smooth, the magnetic field diffuses or slips through the plasma without reconnection;

(b) but if \({\mathbf{u}}\) is singular, then 2D reconnection occurs.

In 2D, therefore, one can either have slippage of the magnetic field, or reconnection (at an X-type null point), or destruction or generation of magnetic flux (at an O-type null point).

3D reconnection, however, has a completely different nature to 2D reconnection (Sect. 4.4). For example, in 2D, reconnection takes place at an X-point, with the field lines slipping through the plasma in the diffusion region in a manner described by the flux velocity and changing their connections only at the X-point. However, none of these properties hold in 3D, where reconnection may take place at a null or at a separator or quasi-separator (or hyperbolic flux tube), and in the diffusion region field lines continually change their connections. Also, the concept of a flux velocity has to be rethought, since a single flux velocity no longer exists (Sect. 3.3.3, Sect. 4.4).

All of the above concepts are developed in detail in Priest (2014) and briefly in the following sections. However, we begin in the next two sections by discussing important aspects of magnetic field structure in two and three dimensions.

2 Topological and geometrical features of magnetic fields

In two dimensions reconnection occurs only at null points (described in Sect. 2.1). These nulls may be pre-existing in the field and undergo local collapse to form a current sheet (Sect. 2.1), or a one-dimensional current sheet may form, within which a 2D null is created when reconnection is initiated. By contrast, in three dimensions, reconnection is not constrained to occur only at null points: several other magnetic field structures may be sites of reconnection, since they are natural locations where current concentrations tend to form and dissipate. These include separatrix surfaces and their intersections in separator curves (Sect. 2.3), which form a topological skeleton or web-like structure in a complex magnetic configuration (Sect. 2.5). Separatrices contribute to the topological structure of a configuration, which may undergo sudden changes in structure, called bifurcations (Sect. 2.4). In addition, non-topological (i.e., geometrical) features, called quasi-separatrix layers (QSLs), can be an important part of the geometry of a magnetic field and form a quasi-skeleton. They intersect in quasi-separators or hyperbolic flux tubes (HFTs), where strong currents may also accumulate (Sect. 2.6). The term structural skeleton has been suggested by Titov to refer to the sum of the topological and quasi-skeleton (i.e., both the separatrices and QSLs) and it may be best identified by Titov’s Q-factor (Sect. 2.6) (Titov 2007; Titov et al. 2009). For an in-depth account of magnetic topology which complements this review, see Longcope (2005).

2.1 Null points in two dimensions

Special locations in a magnetic configuration where the field vanishes are called neutral points or null points. In 2D, they come in two types, X-points or O-points, near which the field lines are hyperbolic or elliptic, respectively (Fig. 2). X-points have a tendency to collapse and form intense sheets of current where reconnection takes place, whereas O-points can be locations of creation or destruction of magnetic flux.

Sufficiently close to a generic magnetic null, the field is dominated by linear terms and has the form

$$\begin{aligned} {\mathbf {B}}=[B_x,B_y] = \frac{B_0}{r_0}[y,\bar{\alpha }^{2} x], \end{aligned}$$

where \(B_0\), \(r_0\) and \(\bar{\alpha }\) are constants.

Fig. 2
figure 2

Examples of 2D null points of (left) O-type and (right) X-type

From \({\mathbf{j}}={{{\varvec{\nabla }}}} \times {\mathbf{B}}/\mu \), the value of the current density (in the z-direction) is

$$\begin{aligned} j_{z} = \frac{B_{0}}{\mu r_{0}} (\bar{\alpha }^{2} - 1). \end{aligned}$$

O-points arise when \(\bar{\alpha }^{2} < 0\), with the particular case \(\bar{\alpha }^{2} = -1\) giving circular field lines (Fig. 2a). X-points arise when \(\bar{\alpha }^{2} > 0\), for which the limiting field lines \(y = \pm \, \bar{\alpha } \, x\), known as separatrices, are inclined at \(\pm \tan ^{-1} \bar{\alpha }\) to the x-axis. The particular case \(\bar{\alpha } = 1\), gives a separatrix angle of \(\textstyle {\frac{1}{2}}\pi \) (Fig. 2b) and makes the current density vanish (Eq. 4).

An X-type neutral point tends to be locally unstable if the sources of the magnetic field are free to move (Dungey 1953). This may be demonstrated by a qualitative physical analysis, a linear analysis or a nonlinear self-similar solution, as detailed in Priest (2014).

An equilibrium X-point (Eq. 3) with \(\bar{\alpha }=1\) has field lines given by \(y^{2} - x^{2} = {\mathrm {const}}\) (Fig. 3a), and is acted on by a Lorentz force (\({\mathbf {j}}\times {\mathbf {B}}\)), which may be split into a magnetic tension force (\({\mathbf{T}} \equiv {\mathbf {B}}\cdot \nabla ){\mathbf {B}}/\mu \)) acting outwards and a magnetic pressure force (\( {\mathbf{P}}\equiv -\nabla (B^2)/(2\mu )\)) acting inwards. Initially, these two balance one another, but, if the field is distorted by keeping the form of Eq. (3) and letting \(\bar{\alpha }^{2}< 1\), it is no longer in equilibrium (Fig. 3b). The equation for the magnetic field lines becomes \( y^{2} - \bar{\alpha }^{2} x^{2} = {\mathrm {const}}, \) with the separatrix field lines \((y= \pm \, \bar{\alpha } x)\) no longer being inclined at \(\textstyle {\frac{1}{2}}\pi \). Along the x-axis, the outwards magnetic tension force now dominates the magnetic pressure due to the increase in field-line curvature, while along the y-axis the inwards magnetic pressure dominates. The equilibrium (3) is unstable, because the Lorentz force tends to make the separatrices close up even more. The resulting increase in \(\bar{\alpha }\) implies that the current density also increases.

Fig. 3
figure 3

(left) The field lines of an X-point in equilibrium, with perpendicular separatrices, such that the magnetic pressure (P) and tension (T) forces acting on a plasma element (shaded) balance one another. (right) A perturbation with uniform current produces a resultant force \(R=T+P\) that acts to make the separatrices close up. Here we show P and T in just the top and left quadrants and R in just the bottom and right quadrants

Note that the instability occurs only if conditions at distant boundaries allow. It cannot take place in, for example, a potential magnetic configuration whose field lines are tied to the boundaries, since such a minimum-energy state would be stable. However, for a more complex non-potential configuration containing extra energy, collapse may be a means of forming strong currents and so dissipating the excess energy. A series of thorough linear analyses of X-type collapse have been undertaken (e.g., Bulanov et al. 1990; Craig and McClymont 1991, 1993; Titov and Priest 1993). They demonstrate that collapse occurs for a wide variety of initial and boundary conditions, provided the perturbation rate is fast enough that dynamic effects are important, and they show surprisingly that magnetic reconnection during the linear regime is fast and scales as 1/(ln \(\eta \)).

2.2 Null points in three dimensions

At a linear 3D null point the magnetic field vanishes, and nearby the field increases linearly with distance from it. The simplest example has magnetic components

$$\begin{aligned} (B_{x}, B_{y}, B_{z}) = (x,y, -2z), \end{aligned}$$

and obeys \({\varvec{\nabla }\cdot {\mathbf {B}}} = 0\) identically. The magnetic field lines satisfy \(dx/B_{x} = dy/B_{y} = dz/B_{z}\), and so their equations are given by \( y=Cx\) and \(z= K/x^{2}, \) where C and K are constant. They give rise to quite a different structure from 2D (see Fig. 4).

Fig. 4
figure 4

The two main features of a 3D null point are the spine field line [using Priest and Titov (1996)’s notation] (or \(\gamma \)-line, using Lau and Finn (1990)’s notation) and the fan surface (or \({\varSigma }\)-surface). For a a proper radial null, the field lines of the fan spread out radially in all directions, while for b an improper radial null, most of them touch one of the directions. c A spiral null point occurs when \(j_{\parallel }\) exceeds a critical value, while d an oblique null point results when \(j_{\perp } \ne 0\) (see Eq. 6)

Two special families of field lines pass through a 3D null point (Fig. 4a), as first discussed in a far-sighted paper by Lau and Finn (1990). The terms spine and fan were later coined for them by Priest and Titov (1996). In the linear null point field of Equation (5), the spine curve is the single field line that approaches or recedes from the origin along the z-axis, with nearby field lines making up two bundles that spread apart as they approach the xy-plane (i.e., the fan surface). When the fan field lines radiate from the null we refer to it as a positive null point, whereas when they converge on the null we call it a negative null point. A positive null has topological degree \(-1\) while a negative null has topological degree \(+1\): they were referred to as “B-type” and “A-type”, respectively, by Lau and Finn (1990).

The above null point (Equation 5) is referred to as a proper radial null, since it has fan field lines that are straight. A broader class of null points has field components

$$\begin{aligned} (B_{x}, B_{y}, B_{z})=(x, ay, -(a+1)z). \end{aligned}$$

When \(a \ne 1\), we have an improper radial null, whose fan field lines are curved, touching the y-axis when \(0<a<1\) and the x-axis when \(a>1\) (Fig. 4b).

A general linear null possesses a magnetic field with nine arbitrary constants. However, these may be reduced to only four independent constants (\(a,b, j_{\parallel }, j_{\perp }\)) by using \({\varvec{\nabla }\cdot {\mathbf {B}}} =0\), normalising and rotating the coordinate axes (Parnell et al. 1996), so that

$$\begin{aligned} \left( \begin{array}{c}B_{x} \\ B_{y} \\ B_{z} \end{array}\right) = \left( \begin{array}{ccc}1 &{} \frac{1 }{ 2}(b-j_{\parallel }) &{} 0 \\ \frac{1 }{ 2} (b+j_{\parallel }) &{} a &{} 0 \\ 0 &{} j_{\perp } &{} -a-1 \end{array}\right) \left( \begin{array}{c} x \\ y \\ z \end{array}\right) . \end{aligned}$$

When the component \((j_{\perp })\) of the current perpendicular to the spine vanishes, the fan surface is normal to the spine: otherwise it is referred to as an oblique null (Fig. 4d). When the component \((j_{\parallel })\) of current parallel to the spine is sufficiently large, the eigenvalues of the matrix in Eq. (6) become complex and it transforms into a spiral null, whose fan field lines spiral into or out of the null (Fig. 4c). The collapse of a linear 3D null point in different ways has been studied, for example, by Bulanov and Olshanetsky (1984), Parnell et al. (1997), and Mellor et al. (2003).

The existence of null points in the solar corona has been discussed by several authors. Twelve hexagonal supergranule cells with sources at their boundaries and centres were modelled by Inverarity and Priest (1999). The distribution of coronal null points due to a random distribution of photospheric sources possesses typically one coronal null for every ten photospheric sources (Schrijver and Title 2002; Longcope et al. 2003). For a field extrapolated from an observed MDI magnetogram, Longcope and Parnell (2009) found one null above a height of 1.5 Mm per 322 Mm\(^{2}\) patch of the quiet Sun, but this has not yet been repeated for higher-resolution SDO/HMI or SUNRISE magnetograms. Freed et al. (2015) searched for magnetic nulls in potential field source-surface (PFSS) extrapolations based on synoptic magnetograms from the Wilcox Solar Observatory, and compared them with observations of coronal emissions from SDO/AIA. Using only 29 harmonics in the expansion they found 582 null points in the extrapolated field between \(1.05\,R_{\odot }\) and \(2.5\,R_{\odot }\). They also discussed whether the presence of nulls can be inferred from coronal observations of hyperbolic X-type shapes in the emission.

2.3 Separatrices and separators

The coronal magnetic field may be modelled as being produced by continuous or discrete photospheric flux sources and sinks. A two-dimensional field in general contains special field lines, called separatrix curves, which divide the plane up into topologically distinct regions, each containing the field lines that start from a particular source and end at a particular sink. Figure 5a,b illustrates the two types of separatrix curve that are possible in 2D, namely, those that originate at X-points and those that touch the boundary in a so-called bald patch (Titov et al. 1993). During 2D reconnection, the field lines are broken and rejoined at an X-point as magnetic flux moves from one topological region to another across the separatrices.

Fig. 5
figure 5

In two dimensions, separatrix curves a may arise from an X-point or b they may touch a boundary. c In three dimensions, an example of the topology of a global potential field extrapolation. Null points are dark blue and red, while separatrix surfaces are pale blue and pink, intersecting in separators (yellow). Images (a) and (b) are reproduced with permission from Priest (2014), copyright by CUP, and (c) from Platten et al. (2014), copyright by ESO

These ideas naturally extend into three dimensions, where surfaces of field lines called separatrix surfaces or separatrices divide the volume into regions of distinct field-line connectivity. When these surfaces intersect one another, they do so in a special field line called a separator, which begins and ends at null points or on the boundary (Figs. 5c, 6). Separators were first considered in a ground-breaking paper by Sweet (1958b) and later analysed by many others (Lau and Finn 1990; Priest and Titov 1996; Longcope and Cowley 1996; Longcope 2005; Parnell et al. 2010). The separatrix surfaces are of two types, namely separatrix fan surfaces, which are the fan surfaces of null points described in the previous section, and separatrix touching surfaces, which touch a boundary in a curve referred to as a bald patch (Fig. 5) (Seehafer 1986; Titov et al. 1993; Bungey et al. 1996; Titov and Démoulin 1999). The concept of a touch curve for the Earth’s magnetic field was proposed by Hide (1979), and the role of bald patches has been considered for solar prominences and flares (Delannée and Aulanier 1999; Aulanier et al. 2000; Aulanier and Schmieder 2002; Schmieder et al. 2001; Pariat et al. 2004). The global topology or so-called skeleton of complex fields due to many sources then comprises a network of separatrix surfaces and their associated null points and separators. Many different configurations are possible, as discussed in Sect. 2.5.

Fig. 6
figure 6

(courtesy of K. Galsgaard)

Magnetic field line structure formed when the fan surfaces of two null points intersect to form a separator line (red) joining the two nulls. For a related movie see Supplementary Information. The movie shows a rotating 3D perspective

Two types of separatrix may be seen in Fig. 7, namely, open separatrices and closed separatrices (Priest 2014). Open separatrix surfaces are bounded by spines and meet the boundary in non-closed curves, so that, even though the field-line mapping has a discontinuity as the separatrix is crossed, all the field lines occupy one region, since any two points in the volume can be joined by a curve that does not cross any separatrices. On the other hand, closed separatrices meet the boundary in closed curves, and so form closed flux surfaces that split the volume into topologically different regions.

Fig. 7
figure 7

Image reproduced with permission from Priest (2014), copyright by CUP

The topological structure of the field near a pair of nulls (\(\hbox {N}_1\) and \(\hbox {N}_2\)), with spines \(\hbox {S}_1\) and \(\hbox {S}_2\) and fans \(\hbox {F}_1\) and \(\hbox {F}_2\). In a the fans of the two nulls are open separatrices that intersect the boundary in non-closed curves (indicated by dotted lines) and intersect each other in a single separator line. This is the generic case for the creation or destruction of a separator by a local bifurcation. In b the fans are closed separatrices that intersect the boundary in closed curves and each other in a double separator forming a closed curve; the spines of the other nulls lie on them but do not bound them: they represent the only field lines that connect to the other nulls.

2.4 Changes in topology and creation of null pairs by a bifurcation

A local bifurcation takes place when the number or nature of null points changes, whereas a global bifurcation in 2D makes the separatrices change their connectivity. In general, a local separator bifurcation may create or destroy isolated null points in pairs (\(\hbox {N}_1\) and \(\hbox {N}_2\), say). The topological structure of the magnetic field during this process is shown in Fig. 7a, where the spines of each null lie in the fans of the other null. The fans (\(\hbox {F}_1\) and \(\hbox {F}_2\)) are bounded by the spines (\(\hbox {S}_2\) and \(\hbox {S}_1\)) of the opposite nulls and form two sheets which intersect in a single separator that links one null to the other (see also Fig. 6). During the creation of a linear null pair, a second-order null appears and then splits into two nulls joined by a separator, as shown. Additional separators joining a pair of nulls can be created by a global bifurcation. For instance, Fig. 7b shows a double separator consisting of a pair of separators, both of which join the two nulls.

Albright (1999) described the way that null points can form in clusters, due to fluctuations in the weak-field region surrounding a null of a “large-scale” field. Thus, another way in which a pair of null points can be created is near a pre-existing null, as shown in Fig. 8. Separators link the new lateral nulls to the central null, and the fan of the central null is an open separatrix (Fig. 8b). A common situation in the corona (Fig. 8c) is to have a separatrix dome possessing three nulls and an open separatrix, which can make up a pseudo-streamer (Titov et al. 2011; Scott et al. 2021).

Fig. 8
figure 8

Image reproduced with permission from Priest (2014), copyright by CUP

a An initial null point (\(\hbox {N}_1\)), which b spawns two new nulls (\(\hbox {N}_2\) and \(\hbox {N}_3\)) by means of a local double separator bifurcation. The resulting topology possesses an open separatrix coming from the fan \(\hbox {F}_2\) of (\(\hbox {N}_2\)) and bounded by spines \(\hbox {S}_1\) and \(\hbox {S}_3\) of the other two nulls (Brown and Priest 2001). c A common situation in the corona where a dome that arches down to the solar surface consists of the fans \(\hbox {F}_1\) and \(\hbox {F}_3\) of two nulls, while an open separatrix expands outwards from a third null and is bounded by the spines \(\hbox {S}_1\) and \(\hbox {S}_3\).

The reason why null points are invariably created in pairs is that the topological degree of the volume must be preserved at all times (unless nulls cross the boundary): nulls contribute either \(+1\) (negative null) or \(-1\) to this topological degree, and so must be created in pairs of opposite sign. Many numerical experiments, such as those on coronal heating or flux emergence, possess a large number of null points that form like beads on a string or chain, joined in multiple ways by short separators (Parnell et al. 2010; Haynes and Parnell 2010). These separators tend to appear as either tiny intracluster separators connecting nulls inside a cluster or as long intercluster separators that join distant nulls or other clusters.

Topological or structural stability refers to a situation where the topological features of a magnetic configuration (comprising its skeleton) are not changed by a small change in the field (Hornig and Schindler 1996). Topological instability, on the other hand, implies that an arbitrary change of the magnetic field causes a topological change. Linear null points are structurally stable in 2D or 3D (Hornig and Schindler 1996), but null lines or null sheets (which comprise curves or surfaces where the field vanishes) are structurally unstable, since they will usually break up into a set of null points when perturbed. In general, null points are structurally unstable when they are degenerate, so that the Jacobian matrix \((D {\mathbf{B}}\)) given by Eq. (6) is singular.

2.5 Skeletons of complex magnetic configurations

To understand the dynamics of complex magnetic fields such as the field of the solar corona, a key question is how to characterise the magnetic field’s structure. The photospheric magnetic field is concentrated by convection into many intense flux tubes, and each corresponding photospheric flux concentration is itself linked in the corona to many other sources by magnetic field lines (e.g., Schrijver and Title 2002). Magnetic nulls, separatrices and separators are potentially important locations for rapid energy conversion, and so together they can reveal important clues to understanding the dynamics of the solar atmosphere. Thus, a powerful way to understand the behaviour of plasmas threaded by such complicated magnetic fields is to construct the skeleton of the field, namely, the set of separatrix surfaces that originate both in the fans of null points and in bald patches and which intersect in separators (see Sect. 2.3).

Early analyses of the different topological magnetic field structures that might be found in the corona made use of the magnetic charge topology approach (see Longcope 2005, for an extensive review). In this model, the photospheric flux concentrations are approximated by magnetic point sources (monopoles). Priest et al. (1997), Brown and Priest (1999) catalogued the different configurations that are possible when three photospheric sources are present, and the bifurcations that occur during transitions between those states. This was extended to four flux sources by Beveridge et al. (2002, 2003). These magnetic charge topology models contain a number of photospheric and coronal null points (and separators joining them). However, when the field is modelled alternatively with continuous flux sources, some of the photospheric nulls are no longer present (Lee and Brown 2020), while some of the separators become quasi-separators (or hyperbolic flux tubes) (Sect. 2.6).

Later, the magnetic skeleton in fields with continuous photospheric flux distributions was analysed in detail, with different characteristic structures identified. Platten et al. (2014) showed that the separatrices associated with coronal magnetic nulls (and the spines of those nulls that typically bound one or more fans as in Figs. 7, 8) can intersect with one another in various different configurations involving both closed and open separatrix structures. They identified new topological features, such as separatrix caves and separatrix tunnels (Fig. 10).

In the solar corona, the skeleton thus possesses several building blocks:

  • separatrix domes, whose photospheric field encloses a region of parasitic polarity, above which lies a coronal null (N) with a fan that closes down to the photosphere (Fig. 9a);

  • separatrix curtains, which spread out from the fans of coronal nulls as open flux sheets that project out into the solar wind, either as closed separatrices or open separatrices (Fig. 9b);

  • bald-patch separatrices, which touch the solar surface at bald patches;

  • separatrix caves, having open separatrix domes with one opening, formed from a dome that does not intersect the photosphere on all sides but is bounded by the spine of an opposite polarity null (Fig. 10);

  • separatrix tunnels, having open separatrix domes with two openings, formed from a dome that does not intersect the photosphere on all sides but is bounded by two spines from opposite polarity nulls (Fig. 10).

Fig. 9
figure 9

Building blocks for coronal magnetic field topology arising from coronal nulls. Separatrix domes are formed by the spreading of the fan of the null point down to the solar surface, while separatrix curtains form when a null-point fan is open into the high corona

Fig. 10
figure 10

Images reproduced with permission from Platten et al. (2014), copyright by ESO

The structure in pink of a separatrix cave (above) and a separatrix tunnel (below). 3D images of the topology are shown (left) together with maps of the locations where the skeleton meets the photosphere (right). Blue represents a separatrix curtain and green a heliospheric current sheet curtain.

2.6 Quasi-skeletons

2.6.1 Quasi-separatrix layers (QSLs) and the squashing factor

As described above, magnetic nulls and separators are potential sites of reconnection in three dimensions. This is in part because they produce discontinuities of the field-line mapping, and so stress tends to accumulate at these structures and drive growth of the electric current (see Sect. 5). However, intense electric currents can also accumulate at locations where the field-line mapping (or connectivity) is continuous, but exhibits steep gradients. Such locations are called quasi-separatrix layers (QSLs), a concept first proposed by Priest and Démoulin (1995), built on an earlier idea of singular field lines (Priest and Forbes 1989) and later improved (Titov et al. 2002) and soon applied extensively to active regions and flares (Démoulin et al. 1996a, 1997b, a; Démoulin 2006; Aulanier et al. 2006). A collection of QSLs of a field is known as a quasi-skeleton.

Consider the mapping of a 2D X-point field (\(B_{x}=x, \ B_{y} = -y\)) from one point \((x_{0}, y_{0})\) on the boundary to another point \((x_{1}, y_{1})\). When \((x_{0}, y_{0})\) crosses a separatrix, there is a discontinuity in the mapping as the point \((x_{1}, y_{1})\) suddenly jumps in location (Fig. 11a).

Fig. 11
figure 11

Image reproduced with permission from Priest and Démoulin (1995), copyright by AGU

a A 2D X-field has a discontinuity in the mapping of footpoints from the top boundary of a square to the side boundary, b but a 3D sheared X-field has a mapping that is continuous from one plane \(z=0\) to another plane \(z=L\).

In 3D, discontinuities in the mapping are also present at separatrix surfaces that spread out from the fans of null points or from bald patches. In the absence of nulls and bald patches, there are no separatrices, and so the mapping from one footpoint to another is continuous (Schindler et al. 1988). Priest and Démoulin (1995), however, discovered that, even when the mapping is continuous, there often exists a remnant structure known as a quasi-separatrix layer, where the gradient of the mapping is much larger than normal. For example, suppose we add a uniform field \(l{\hat{\mathbf{z}}}\) to the 2D X-point field to give

$$\begin{aligned} (B_{x}, B_{y}, B_{z}) = (x,-y,l) \end{aligned}$$

inside a cube of side 1 (\(-\textstyle {\frac{1}{2}}\leqslant x \leqslant \textstyle {\frac{1}{2}},\ -\textstyle {\frac{1}{2}}\leqslant y \leqslant \textstyle {\frac{1}{2}},\ 0 \leqslant z \leqslant 1\)). Suppose \(l \ll 1\), so that \(e^{-1/l}=\epsilon \ll 1\). Then the planes \(x=0\) and \(y=0\) are QSLs, which can be seen as follows. The mapping from the base \(z=0\) to the top \(z=1\) and side \(x=\textstyle {\frac{1}{2}}\) is given by

$$\begin{aligned} x_{1} = x_{0} \, e^{z_{1}/l}, \qquad y_{1} = y_{0} \, e^{-z_{1}/l}. \end{aligned}$$

Thus, as the point A moves a small distance across \(x=0\) from \(x_0=-e^{-1/l}\) to \(x_0=e^{-1/l}\), while \(y_0\) remains constant, so \(x_1\) moves rapidly over a large distance on the top from \(-1/2\) to \(+1/2\), as indicated in Fig. 11b (\(y_1\) remains constant and very much smaller than \(y_0\)).

In order to investigate more generally the quasi-topology of a 3D configuration, the technique is to calculate the mapping of field-line footpoints from one part of a boundary to the other and to determine where the mapping gradient is large (Priest and Démoulin 1995). For example, follow Titov et al. (2002) in considering a typical solar active region with field lines joining photospheric domains of positive and negative polarity (Fig. 12). Set up Cartesian coordinates with \(z=0\) representing the photosphere and suppose the two photospheric footpoints of a given field line have coordinates \((x_{+}, y_{+})\) and \((x_{-}, y_{-})\). The mappings are represented by vector functions \( X_{-}(x_{+}, y_{+})\) and \(Y_{-}(x_{+}, y_{+})\) for the mapping in one direction, as well as \( X_{+}(x_{-}, y_{-})\) and \(Y_{+}(x_{-}, y_{-})\) for the mapping in the opposite direction. Priest and Démoulin (1995) had suggested that the location of QSLs can be found from the condition \(N_{\pm } \gg 1\), where \(N_{\pm }\) are the norms of the footpoint mapping matrices, defined by

$$\begin{aligned} N_{\pm } \equiv N(x_{\pm }, y_{\pm }) = \left[ \left( \frac{\partial X_{\mp } }{ \partial x_{\pm }} \right) ^{2} + \left( \frac{\partial X_{\mp } }{\partial y_{\pm }} \right) ^{2} + \left( \frac{\partial Y_{\mp } }{ \partial x_{\pm }} \right) ^{2} + \left( \frac{\partial Y_{\mp } }{\partial y_{\pm }} \right) ^{2} \right] ^{1/2}. \end{aligned}$$

Applying this condition for the location of a QSL to magnetic fields in active regions worked well (e.g., Démoulin et al. 1997a), but Titov et al. (2002) and Titov (2007) realised that it could be improved. They suggested instead that N be normalised to give the so-called squashing factor (Q) and imposed the condition \(Q \gg 2\) to identify QSLs, where Q is defined to be either

$$\begin{aligned} Q_{+} = \frac{-N_{+}^{2}}{B_{z+}/B_{z-}} \qquad \mathrm{or} \qquad Q_{-} = \frac{-N_{-}^{2}}{B_{z-}/B_{z+}}. \end{aligned}$$

When a circle of footpoints is mapped along field lines to give an ellipse, its aspect ratio is given by \(Q/2 +\sqrt{Q^2/4-1}\), which tends to Q when \(Q \gg 2\). Thus, Q represents the degree of squashing of an infinitesimal flux tube, which becomes a thin layer-like flux tube when \(Q \gg 2\). The basic properties of Q are as follows:

  1. (i)

    Q is independent of the direction of the mapping so \({Q}_+\)= \({Q}_-\);

  2. (ii)

    \(Q \rightarrow \infty \) at a separatrix surface;

  3. (iii)

    \(Q\gg 2\) at a quasi-separatrix layer;

  4. (iv)

    Maps of Q identify the locations (i.e., both separatrices and QSLs) where large current densities may accumulate, provided appropriate velocities are present, and therefore where reconnection has the potential to occur.

Titov stresses that a QSL is a geometric rather than a topological feature and also emphasizes its importance for current sheet formation by stagnation-point flows (Cowley et al. 1997; van Ballegooijen 1985; Mikić et al. 1989; Longcope and Strauss 1994; Galsgaard and Nordlund 1996).

Fig. 12
figure 12

Image reproduced with permission from Titov et al. (2002), copyright by AGU

The photospheric plane and magnetic field lines connecting positive and negative polarities from \((x_{+}, y_{+})\) to \((x_{-}, y_{-})\), which are separated by the polarity inversion line (IL).

Later, Titov (2007) derived a covariant form for the squashing factor (Q) which enabled him to diagnose the presence of QSLs in closed and open configurations with arbitrary boundary shapes. He also showed that the perpendicular squashing factor (\(Q_{\perp }\)) is superior to Q, since it eliminates the projection effect that is present in field lines that nearly touch a boundary. Furthermore, Titov et al. (2009) introduced the concept of slip-forth (\(Q_{sf}\)) and slip-back squashing factors (\(Q_{sb}\)), which enable the identification of flux tubes that have either just reconnected or are about to be reconnected (Sect. 12.1).

2.6.2 Examples of QSLs

Titov (2007) described two examples of simple configurations for modeling solar flares. The first is a potential quadrupole configuration (Sweet 1969; Baum and Bratenahl 1980; Gorbachev and Somov 1988), which possesses a separator joining two nulls when the photospheric magnetic flux is concentrated in discrete patches. However, when the photospheric flux is distributed smoothly, the nulls and separator disappear, but a quasi-separator (or hyperbolic flux tube) remains, as shown in Fig. 13.

Fig. 13
figure 13

Image reproduced with permission from Titov et al. (2002), copyright by AGU

A potential quadrupole configuration, showing the photospheric distributions of a the squashing degree Q superimposed on a few iso-contours of the corresponding magnetogram and b half of the magnetic flux surface \(Q=100\) demonstrating the shape of the mid cross-section of the HFT.

The quasi-separator represents the intersection of two quasi-separatrix layers (QSLs), and the region around the quasi-separator is known as a Hyperbolic Flux Tube (HFT) (Titov et al. 2002). The Hyperbolic Flux Tube is bounded by the magnetic flux surface \(Q={\mathrm {const}} \gg 2\) and has a shape that continuously transforms along the tube from a narrow flattened tube to a cross and then to a second orthogonal narrow flattened tube at the other end, as follows (Titov et al. 2002):

figure a

The second example is a model for a twisted flux tube, which also contains a pair of QSLs and a quasi-separator (or hyperbolic flux tube) (Démoulin et al. 1996b). Titov and Démoulin (1999) suggested an approximate, cylindrically symmetric equilibrium for a thin force-free toroidal flux rope with a net current I, major radius R and minor radius a (Fig. 14). The symmetry axis of the flux rope lies below the photospheric plane \(z=0\) at a depth d. There is a balance between the outward radial \({\mathbf {j}}\times {\mathbf {B}}\) self-force of the flux rope and the field of two magnetic charges of opposite sign located on that symmetry axis below the photosphere at distances \(\pm L\) from the torus plane. The field outside the torus is current-free and contains a magnetic X-line.

Fig. 14
figure 14

Image reproduced with permission from Titov and Démoulin (1999), copyright by ESO

The Titov–Démoulin model of a circular force-free flux rope (left) with a net current I, embedded in a potential background field produced by two subphotospheric magnetic charges (\(-q\), q) and a line current (\(I_0\)). The resulting coronal field has a photospheric magnetogram (right) that resembles that of a typical solar active region. The solid and dashed curves represent positive and negative iso-contours of \(B_{z}\), respectively.

A line current \(I_{0}\) is added along the symmetry axis and creates a toroidal field component, which turns the environment of the X-line into an HFT, as can be seen in the photospheric distribution of Q in Fig. 15a (Titov 2007).

Fig. 15
figure 15

Image reproduced with permission from Titov (2007), copyright by AAS

For the twisted flux tube configuration of Fig. 14a the photospheric distribution of the squashing degree Q together with magnetogram iso-contours, and b a cut through the HFT by a midplane with its cross-section and footprint shown in black and white, respectively.

The most intense squashing occurs in very thin QSLs whose footprints have the shape of narrow fishhook-like strips. Fig. 15b shows the flux surface \(Q=100\) of the HFT. The variation of its cross-section is similar to that of the first example, except that the whole structure is also twisted.

2.7 Global topology of coronal magnetic fields and methods of analysis

Several complementary tools have been developed in order to find the skeleton or quasi-skeleton of a magnetic configuration. For the magnetic skeleton, the first step is to locate the magnetic nulls, for which a common, robust way is the trilinear method described by Haynes and Parnell (2007). Other techniques for finding magnetic nulls exist, depending on the type of data, and these are compared by Olshevsky et al. (2020). From the nulls, the spine and fan field lines can be followed. Finding separator lines is challenging since the field lines tend to diverge away from them, but subtle techniques have been developed by Haynes and Parnell (2010).

Platten et al. (2014) used these methods to find the magnetic skeleton of the global corona; they noted incredible complexity, especially as the resolution of the magnetogram used for coronal field extrapolation is increased. In general, as the magnetogram resolution is increased, many more nulls appear, usually at low altitudes. Generally speaking, at solar minimum the apex of the helmet streamer separatrix remains relatively close to the equator, and open and closed separatrix fans are found in abundance at low latitudes (Fig. 16a). On the other hand, at solar maximum the complexity tends to be greater, with separatrices of different types extending up to the source surface at all latitudes (Fig. 16b). Platten et al. (2014) made a statistical analysis of the topological complexity during the cycle. They found that the number of nulls is anti-correlated with the sunspot number (and as a result so too is the number of separators), which can be understood from the fact more of the photosphere is covered by mixed polarity at solar minimum. On the other hand the nulls and associated separators are located higher in the corona at solar maximum. Again there is an intuitive explanation, namely, that null points tend to form at heights on the order of the separation of the photospheric flux patches responsible for their presence, and these flux patches are large at solar maximum. While this and many previous studies (see Sect. 2.5) focussed on potential fields, Edwards et al. (2015) analysed the topological skeleton of a force-free model for the coronal field. They showed that there are substantial differences with the topology of the equivalent potential field models: for example, open field domains (coronal holes) that are present in one model can be absent in the other. It is also worth noting that, while the existence of nulls is relatively robust to the extrapolation method, the details of the null spine-fan structure can be dependent on the force-free extrapolation method (Metcalf et al. 2008). It is clear that extension beyond potential field models is an important future avenue of research.

Fig. 16
figure 16

Image reproduced with permission from Platten et al. (2014), copyright by ESO

Magnetic skeleton with positive (green) and negative (blue) field lines, as calculated by Platten et al. (2014) at a solar minimum and b solar maximum. Positive and negative polarity open-field regions are labelled \(\hbox {OF}_{P1}\), \(\hbox {OF}_{P2}\), ...and \(\hbox {OF}_{N1}\), \(\hbox {OF}_{N1}\), ...respectively. Topological structures include separators (thick yellow), and separatrix surfaces (green and blue) traced from nulls and from the null lines (thick green and blue) on the outer boundary at the base of the heliospheric current sheet (HCS).

One disadvantage of calculating the magnetic skeleton is that QSLs and HFTs are not included, leaving an incomplete picture of potential sites for reconnection. By contrast, the squashing factor (Q) reveals the locations of both separatrices and QSLs. However, it does not distinguish between them, except for the fact that separatrices appear generally as thinner structures in practice (e.g., Titov 2007; Titov et al. 2012; Masson et al. 2012). This is because separatrices correspond to surfaces at which \(Q\rightarrow \infty \), whereas the finite resolution of such calculations means that they deal only with large, finite Q values. A recent advance which allows fast, accurate calculation of Q is the qslsquasher code developed by Tassev and Savcheva (2017). It allows for accurate calculations of Q that are not susceptible to noise, even for large integration steps along the field line, so that Q can be calculated much more quickly than before (for a given spatial resolution).

Thus, in order to obtain a full and accurate picture of the important magnetic field structures for reconnection, both the skeleton and the squashing factor should be obtained. Recently, Scott et al. (2018, 2019) have used global calculations of Q to segment the coronal volume into magnetic flux domains bounded by surfaces of high Q, and have additionally calculated the magnetic nulls. Their motivation was to explore the nature of structures bounding open and closed magnetic flux in the corona: it is proposed that “interchange” reconnection—i.e., reconnection between closed and open magnetic field lines—may be important for structuring the solar wind (Fisk et al. 1998; Crooker et al. 2002; Antiochos et al. 2011)—see Sect. 15.2. They discovered that, in a survey of potential fields, approximately half of these “S-web” structures contain magnetic nulls, while the other half do not (Scott et al. 2019). This highlights the importance of determining the locations of both the QSLs and the nulls and separatrices.

3 Conservation of magnetic flux or field lines

Central to understanding magnetic reconnection in three dimensions are the concepts of flux and field-line conservation, which are much more subtle than normally appreciated. For an ideal MHD plasma, the situation is straightforward, since both flux and field lines are conserved, and the plasma velocity is identical with the flux velocity and the field-line velocity (Sect. 3.1). In nonideal MHD, however, flux and field-line conservation are no longer equivalent (Hornig and Schindler 1996), and neither flux velocity nor field-line velocity is unique (Sect. 3.2). Hornig (1997) has developed a more general concept of Electromagnetic Flux Conservation, of which magnetic flux conservation is just a subclass [see Priest (2014)], but here we focus just on the concepts of flux and field-line conservation, following Hornig (1997, 2001).

3.1 Conservation in an ideal plasma

When the magnetic Reynolds number is very large (\(R_{m} \gg 1\)), the plasma behaves in an ideal manner. The induction equation for magnetic field evolution and Ohm’s law reduce to

$$\begin{aligned} \frac{\partial {\mathbf{B}} }{ \partial t} = {\varvec{\nabla }\times ({\mathbf{v}} \times {\mathbf {B}})} \ \ \ \mathrm{and} \ \ \ {{{\mathbf{E}} + {\mathbf{v}} \times {\mathbf{B}} =0}}. \end{aligned}$$

It may then be shown that magnetic flux, magnetic field lines and magnetic topology are all conserved, and that the components perpendicular to the magnetic field of plasma velocity (\({\mathbf {v}}_{\perp }\)), flux velocity (\({\mathbf {w}}_{\perp }\)) and field-line velocity (\({\mathbf {w}}_{L\perp }\)) are the same, namely,

$$\begin{aligned} {{\mathbf {v}}}_{\perp } ={{\mathbf {w}}}_{\perp } = {{\mathbf {w}}}_{L\perp }= \frac{{\mathbf {E}}\times {\mathbf {B}}}{B^{2}}. \end{aligned}$$

For Magnetic Flux Conservation, the magnetic flux (\(\int _{S} {{{\mathbf{B}} \cdot {\mathbf{dS}}}}\)) through any surface composed of plasma elements is fixed (Fig. 17). Magnetic Field Line Conservation, on the other hand, implies that two plasma elements that are initially linked by a magnetic field line, will continue to be so at later times (Fig. 18).

Fig. 17
figure 17

Magnetic flux conservation: the flux through a curve \(C_{1}\) at time \(t_{1}\) remains constant when it is distorted into a curve \(C_{2}\) at time \(t_{2}\) by a plasma motion

Fig. 18
figure 18

Magnetic field-line conservation: plasma elements (red, black, green circles) that lie on the same field line at an initial time (\(t_{1}\)) will continue to lie on a single field line at later times (\(t_{2}\))

3.2 Conservation in a non-ideal plasma

When the plasma is not ideal, we assume that Ohm’s law takes the form

$$\begin{aligned} {{{\mathbf{E}} + {\mathbf{v}} \times {\mathbf{B}} = {\mathbf{N}}}}, \end{aligned}$$

where the term \({\mathbf{N}}\) on the right is any general non-ideal term due to for instance collisions, fluctuations, particle inertia or classical resistivity; in the case of classical resistivity, it is written \({\mathbf{N}} = \eta {{{\varvec{\nabla }}}} \times {\mathbf {B}}\). We shall find that flux and field-line conservation depend on the form of \({\mathbf{N}}\).

3.2.1 Magnetic flux conservation in a non-ideal plasma

The concept of flux transport can be extended to include non-ideal plasmas, although the velocity of this transport is not unique. If a magnetic flux velocity \(({\mathbf{w}})\) exists with the same flux-preserving property as in ideal MHD, namely, one for which

$$\begin{aligned} \frac{\partial {\mathbf{B}} }{\partial t} = {\varvec{\nabla }} \times ({\mathbf{w}} \times {\mathbf{B}}), \end{aligned}$$

then we say that magnetic field evolution satisfying Eq. (10) is flux-preserving. For an Ohm’s Law of the form (9), Faraday’s equation (\(\partial {\mathbf {B}}/ \partial t=-{{{\varvec{\nabla }}}} \times {\mathbf {E}}\)) implies that

$$\begin{aligned} \frac{\partial {{\mathbf {B}}} }{\partial t} = {{{\varvec{\nabla }}}} \times ({{\mathbf {v}}} \times {{\mathbf {B}}} - {\mathbf{N}}). \end{aligned}$$

Comparing Eq. (10) with Eq. (11), we see that a flux velocity \(({\mathbf{w}})\) exists—and so Magnetic Flux is Conserved—provided the nonideal term in Eq. (9) can be written in the form

$$\begin{aligned} {\mathbf{N }} = {\mathbf {u}}\times {\mathbf {B}}+ {\varvec{\nabla }}{\varPhi }, \end{aligned}$$

where \({\varPhi }\) is a potential and the difference (\({\mathbf {u}}\equiv {\mathbf {v}}- {\mathbf {w}}\)) between the plasma and flux velocities is the slippage velocity. Since \({\mathbf {u}}\times {\mathbf {B}}\) is perpendicular to \({\mathbf {B}}\), it can be seen from Eq. (9) that it is the \({{\varvec{\nabla }}} {\varPhi }\) term that can produce a component (\(E_{\parallel }\)) of \({\mathbf {E}}\) along the magnetic field, which is essential for 3D reconnection (Sect. 4.5). Furthermore, Ohm’s law can then be written in terms of the flux velocity as

$$\begin{aligned} {\mathbf{E}} + {\mathbf {w}}\times {\mathbf {B}}= {\varvec{\nabla }}{\varPhi }. \end{aligned}$$

In addition, magnetic flux is conserved provided \({\mathbf{N}}\) satisfies

$$\begin{aligned} {{{\varvec{\nabla }}}} \times {\mathbf{N}} =0. \end{aligned}$$

3.2.2 Magnetic flux velocity

Since displacements along the magnetic field are arbitrary when considering transport of flux, we may set them to zero, so that \(({\mathbf{w}} - {\mathbf{v}}) \cdot {\mathbf{B}} = 0\). Then, useful information about the nature of \({\mathbf {w}}\) can be found by taking the vector and scalar products of Eq. (13) with \({\mathbf {B}}\). First of all, taking vector products of the two forms (9) and (13) of Ohm’s law with \({\mathbf {B}}\) and subtracting them implies that the flux velocity may be written

$$\begin{aligned} {\mathbf {w}}= {\mathbf {v}}+ \frac{({\mathbf{N}} - {\varvec{\nabla }} {\varPhi }) \times {\mathbf {B}}}{ B^{2}}, \end{aligned}$$

which implies that \(({\mathbf{w}})\) may become singular at a null point (where \(B=0\)).

Next, take the scalar product of Eq. (13) with \({\mathbf {B}}\) to give

$$\begin{aligned} {\mathbf {B}}\cdot {\varvec{\nabla }}{\varPhi } = {\mathbf{E}} \cdot {\mathbf {B}}, \end{aligned}$$

which, provided nulls or boundary conditions do not lead to difficulty, may be integrated along a field line \({\mathbf {B}}\) from an arbitrary value \({\varPhi }_{0}({\mathbf {r}}_{0}, t)\) at some reference surface \(({\mathbf {r}}_{0})\), say, to give

$$\begin{aligned} {\varPhi } ({\mathbf {r}}, t) = \int ^{{\mathbf {r}}}_{{\mathbf {r}}_{0}} {{{\mathbf{E}} \cdot {\mathbf{ds}}}} + {\varPhi }_{0}({\mathbf {r}}_{0}, t). \end{aligned}$$

Since \({\varPhi }_{0}\) is arbitrary, both \({\varPhi }({\mathbf {r}},t)\) and \(({\mathbf {w}})\) in Eq. (15) are not unique.

When \(E_{\parallel } = 0\) there is no 3D reconnection, but there is magnetic diffusion with a slippage velocity given for a resistive Ohm’s law by Eq. (15) as

$$\begin{aligned} {\mathbf {u}}\equiv {\mathbf{v}} - {\mathbf{w}} = - \frac{{\mathbf{j}} \times {\mathbf{B}} }{ \sigma B^{2}}. \end{aligned}$$

In order to avoid \({\varPhi }\) becoming multiple-valued as one integrates around a closed field line, a necessary condition for (16) to be valid is that

$$\begin{aligned} {\varPhi } = - \oint {{{\mathbf{E}} \cdot {\mathbf{dl}}}} = 0 \end{aligned}$$

around such a field line. When the magnetic field is steady (so that \({{{\varvec{\nabla }}}} \times {\mathbf{E}}=0\)), this is always satisfied, but, when it is unsteady, it may fail (Hornig 2001).

3.2.3 Magnetic field line conservation in a non-ideal plasma

Whenever magnetic flux is conserved, it turns out that magnetic field lines are also conserved. By contrast, there are many evolutions in a non-ideal plasma that conserve field lines but do not conserve flux. In a non-ideal plasma, magnetic flux is conserved if Eq. (10) holds, whereas magnetic field lines are conserved if

$$\begin{aligned} \frac{\partial {\mathbf{B}} }{ \partial t} = {{\varvec{\nabla }}} \times (\mathbf{w}_{L} {{\times \mathbf{B}) + \lambda }}_{L} {\mathbf{B}}, \end{aligned}$$

so that, by comparing with Eq. (11), \({\mathbf{N}}\) must have the property that \({{{\varvec{\nabla }}}} \times {\mathbf{N}}\) is parallel to \({\mathbf{B}}\), i.e., field lines are conserved if

$$\begin{aligned} {\mathbf{B}} \times ({{{\varvec{\nabla }}}} \times {\mathbf{N}}) =0. \end{aligned}$$

In Eq. (17), \({\mathbf{w}}_{L}\) is a magnetic field-line velocity and \(\lambda \) is a scalar function of position, and so, by choosing \(\lambda = \lambda _{L}+{{{\varvec{\nabla }}}} \cdot {\mathbf{w}}_{L}\) and using \({{{\varvec{\nabla }}}} \cdot {\mathbf {B}}=0\), Eq. (17) may be rewritten

$$\begin{aligned} \frac{\partial {\mathbf {B}}}{\partial t}+ ({\mathbf{w}}_{L} \cdot {\varvec{\nabla }}) {\mathbf{B}} - ({\mathbf{B}} \cdot {\varvec{\nabla }}) {\mathbf{w}}_{L} = \lambda {\mathbf{B}}. \end{aligned}$$

Thus, if we choose \(\lambda _{L}=0\) and \({\mathbf{w}}_{L}={\mathbf{w}}\), Eq. (17) for field-line conservation reduces to Eq. (10) for flux conservation. In other words, when flux is conserved, field lines are conserved too. On the other hand, when \(\lambda _{L} \ne 0\), field lines are conserved but flux is not.

3.2.4 Magnetic field line velocity

When the plasma is not ideal, the field-line velocity component \(({\mathbf{w}}_{L\perp })\) can be defined uniquely if and only if Ohm’s Law may be written as

$$\begin{aligned} {\mathbf{E + w}}_{L} {{\times {\mathbf{B}} = {\mathbf{a}}}}, \end{aligned}$$

with \( {{\varvec{\nabla }} \times {\mathbf{a}}} = -\lambda _{L} {\mathbf{B}}, \) in which case Eq. (19) and Eq. (8) imply Eq. (17). This definition is field-line preserving but is only flux preserving when \({{{{\varvec{\nabla }}}} \times {\mathbf {a}}} = {\mathbf{0}}\), so that \({{\mathbf{a}} = {\varvec{\nabla }} {\varPhi }}\) and \(\lambda _{L} =0\).

When the plasma is ideal, it is always possible to write Eq. (19), and so the perpendicular component of the field-line velocity is

$$\begin{aligned} {\mathbf{w}}_{L\perp } = \frac{{\mathbf{E}} \times {\mathbf{B}}}{B^2}, \end{aligned}$$

but it is also possible when \({{{\mathbf{E}} \cdot {\mathbf{B}}}} = 0\). More generally, when Eq. (19) holds, the field-line velocity is

$$\begin{aligned} {\mathbf{w}}_{L\perp } = \frac{({{{\mathbf{E - a}}) \times {\mathbf{B}}}} }{ B^{2}}. \end{aligned}$$

However, this form is not unique, since the magnetic field is unchanged if we replace \({\mathbf{a}}\) by \({\mathbf{a}}^{\prime } = {{{\mathbf{a}} + {\varvec{\nabla }}}} {\varPsi }^{*}\), with \(({{{\mathbf{B}} \cdot {\varvec{\nabla }}})} {\varPsi }^{*} = 0\), such that \({\mathbf{B}} \cdot \)(19) does not change, but the field-line velocity becomes instead

$$\begin{aligned} {\mathbf{w}}_{L\perp }^{\prime } =\frac{({\mathbf{E - a}}^{\prime }) \times {\mathbf{B}} }{B^{2}}. \end{aligned}$$

The concepts of flux and field line velocity, and their uniqueness or otherwise, are invaluable when considering the properties of 3D reconnection (Sect. 4).

3.3 Magnetic diffusion and field-line motion

To complete our discussion of field line motion we consider the case of pure diffusion (with plasma velocity \({\mathbf {v}}=0\)). Wilmot-Smith et al. (2005) presented examples of magnetic diffusion in 1D (Sect. 3.3.1), 2D (Sect. 3.3.2) and 3D (Sect. 3.3.3). Often, diffusion can be described with the help of a magnetic flux velocity, but such a velocity is usually non-unique, so that the field lines may be said to move in several different ways. For straight magnetic field lines with diffusion in a current sheet, the magnetic field behaves as if the flux disappears either at a current sheet and/or at infinity. In a similar manner, circular field lines diffuse as if the magnetic field moves either towards the O-type neutral line and/or towards infinity. In 3D, although magnetic field lines can always be defined at any time, the decay of a field cannot necessarily be described in terms of the motion of magnetic field lines, since it is not always possible to define a flux velocity. Instead, it may be possible to describe the field behaviour in terms of a dual flux velocity (Sect. 3.3.3, Sect. 4.4). For more details on these ideas, see Wilmot-Smith et al. (2005).

Consider resistive diffusion of a magnetic field with uniform magnetic diffusivity (\(\eta \)), for which \({\mathbf {E}}=\eta {{{\varvec{\nabla }}}} \times {\mathbf {B}}\) and the induction equation becomes

$$\begin{aligned} \frac{\partial {\mathbf{B}}}{\partial t}={{{\varvec{\nabla }}}} \times (\eta {{{\varvec{\nabla }}}} \times {\mathbf {B}}) = \eta \nabla ^{2}{\mathbf{B}}, \end{aligned}$$

while the existence of a magnetic flux velocity (or flux transporting velocity) \(({\mathbf{w}})\) requires Eq. (10). 1D, 2D, and 3D scenarios are considered in turn below.

3.3.1 Diffusion of a magnetic field with straight field lines

When a 1D magnetic field (\({\mathbf{B}}=B(x,t)\hat{{\mathbf{y}}}\)) diffuses, magnetic field lines can disappear either at a neutral sheet or at the boundary, and the diffusion equation (20) reduces to

$$\begin{aligned} \frac{\partial B}{\partial t}=\eta \frac{\partial ^{2}B}{\partial x^{2}}. \end{aligned}$$

Consider, for example, a magnetic field whose value is held fixed at two points \((\pm \ell )\) with \(B(\ell ,t)=-B(-\ell ,t)=B_{0},\) and that initially has a step profile with \(B(x,0)=B_{0}\) for \(x>0\), and \(B(x,0)= -B_{0}\) for \(x<0\), representing an infinitesimally thin current sheet. The solution to the diffusion equation is

$$\begin{aligned} B(x,t)=B_{0}\frac{x}{\ell }+\frac{2B_{0}}{\pi }\sum _{n=1}^{\infty } \frac{1}{n}\exp {\left( -n^{2}\pi ^{2}\eta t/\ell ^{2}\right) }\sin \left( \frac{n\pi x}{\ell }\right) , \end{aligned}$$

and the resulting magnetic field diffuses away very rapidly towards the steady-state solution, \(B(x)=B_{0}x/\ell \). In terms of energy, a decrease in magnetic energy is accounted for by Ohmic heating \((j^{2}/\sigma )\) and an outwards Poynting flux \({\mathbf{E}}\times {\mathbf{B}}/\mu =-(\eta /\mu )\partial B/\partial x \ \hat{\mathbf{x}}\) into the boundaries \(x=\pm \ell \). In the final steady state, the ohmic heating (\(j^{2}/\sigma =(\eta /\mu )(B_{0}/\ell )^{2}\) per unit length) is provided by a continual inflow of energy through the boundaries.

In this case, a flux velocity \(({\mathbf{w}}=w\hat{\mathbf{x}})\) exists and Eq. (13) becomes

$$\begin{aligned} \eta \frac{\partial B}{\partial x}+w B=E_{0}(t), \end{aligned}$$

with solution

$$\begin{aligned} w=-\frac{\eta }{B}\frac{\partial B}{\partial x}+\frac{E_{0}}{B}, \end{aligned}$$

where \(E_{0}(t)\) is an arbitrary function representing a nonuniqueness in the form of the flux velocity.

There are several physically reasonable ways of choosing \(E_{0}\), one of which is \(E_{0}=0\), so that \({\mathbf{w}}={\mathbf{E}}\times {\mathbf{B}}/B^{2}\) and \({\mathbf{w}}\) is then a flux velocity associated with the energy flow. In this case, the field lines are initially stationary (except at the origin) and later move towards the origin with a singular velocity at \(x=0\). The field is, therefore, evolving as if the field lines are moving towards the origin and disappearing (or “annihilating”) there at a neutral sheet. As time increases, the flux velocity increases everywhere in magnitude towards its steady-state value.

3.3.2 Diffusion of a magnetic field with circular field lines

Consider next diffusion of a field \(B(r,t){\hat{\varvec{\theta }}}\) where \(B=-\partial A/\partial r\), for which the diffusive limit of the induction equation with uniform diffusivity in terms of the flux function (A(rt)) becomes

$$\begin{aligned} \frac{\partial A}{\partial t}=\eta \left( \frac{\partial ^{2}A}{\partial r^{2}}+\frac{1}{r}\frac{\partial A}{\partial r}\right) . \end{aligned}$$

For diffusion of an isolated circular flux tube of flux \(F_{0}\) at radius a with an initial field \(B(r,0)=F_{0}\delta (r-a)\) and flux \(A(r,0)= 0\) for \(r<a\), and \(A(r,0)= -F_{0}\) for \(r>a\), the solution is, in terms of the Bessel function \(\hbox {I}_{0}\),

$$\begin{aligned} A(r,t)=-\frac{F_{0}}{2\eta t}\int _{a}^{\infty }s e^{-(s^{2}+r^{2}/(4\eta t)}\mathrm{I}_{0}\left( \frac{rs}{2\eta t}\right) \mathrm{d}s, \end{aligned}$$

for which the maximum field strength decreases in time, while the flux spreads outwards.

The resulting total flux is

$$\begin{aligned} A(0,t)-A(\infty ,t)=F_{0}(1-e^{-a^{2}/(4\eta t)}), \end{aligned}$$

which decays away from an initial value of \(F_{0}\) to zero with a time-scale of \(a^{2}/(4\eta )\). The corresponding radial flux velocity is

$$\begin{aligned} w=\frac{1}{B}\left( E_{0}-\eta \frac{\partial B}{\partial r}\right) , \end{aligned}$$

where \(E_{0}(t)\) is an arbitrary function of time, which can again be chosen in a variety of ways. For example, if \(E_{0}(t)\) is chosen to make w vanish at infinity, the field lines would be disappearing at the O-point. This surprising fact that field lines can disappear at O-points in 2D makes one wonder whether they can disappear in 3D.

3.3.3 Magnetic field diffusion in three dimensions

If a closed magnetic field line C exists enclosing a surface S, then the rate of change of magnetic flux through S is

$$\begin{aligned} \frac{d}{dt}\int \limits _{S}{\mathbf{B}} \cdot {\mathrm{d}}{\mathbf{S}}=-\int \limits _S{\varvec{\nabla }}\times {\mathbf{E}} \cdot {\mathrm{d}}{{\mathbf{S}}}=-\int \limits _{C}{\mathbf{E}} \cdot {{\mathrm{d}}{\mathbf{l}}}. \end{aligned}$$

If, further, a flux velocity \(({\mathbf{w}})\) exists, then there is a function \({\varPhi }\) such that Eq. (13) holds, where \({\mathbf{E}}=\eta {\varvec{\nabla }}\times {\mathbf{B}}\), which implies, since \({\mathbf {w}}\times {\mathbf {B}}\) is perpendicular to \(d{\mathbf{l}}\) on C, that

$$\begin{aligned} \int \limits _{C}{\mathbf{E}}\cdot {\mathrm{d}}{\mathbf{l}}=\int \limits _{C}{\varvec{\nabla }}{\varPhi }\cdot {\mathrm{d}}{{\mathbf{l}}}=0, \end{aligned}$$

so that the flux through C does not change in time. Thus, if the flux through a closed field line does indeed change in time, Eq. (13) cannot hold and no flux velocity \(({\mathbf{w}})\) exists.

As an example, consider the diffusion of a linear force-free field satisfying \({\varvec{\nabla }}\times {\mathbf{B}}=\alpha _{0}{\mathbf{B}},\) where \(\alpha _{0}\) is constant. The diffusive induction Eq. (20) reduces to

$$\begin{aligned} \frac{\partial {\mathbf{B}}}{\partial t}=-\eta \alpha _{0}^{2}{\mathbf{B}} \qquad {\mathrm{with\ solution}} \qquad {\mathbf{B}}(x,y,z,t)={\mathbf{B}}_{0}(r,\theta ,\phi )e^{-\eta \alpha _{0}^{2}t}, \end{aligned}$$

where \({\mathbf{B}}_{0}(r,\theta ,\phi )\) is the initial state. As a particular case, consider the lowest-order axisymmetric, linear-force-free field in a sphere of radius a, as sketched in Fig. 19, which has flux function \(A=r^{1/2}\mathrm{J}_{3/2}(\alpha _{0}r)\sin ^{2}\theta \) with \(\alpha _{0}a\) \(\approx 4.49\) as the first zero of the Bessel function \(\mathrm{J}_{3/2}\).

This field possesses a closed field line (C) in the equatorial plane (\(\theta = \textstyle {\frac{1}{2}}\pi \)) at the location \(\alpha _{0} r=2.46\) of the first maximum of \(\partial A/\partial r\). Within C the poloidal flux decreases in time, and so no single flux velocity exists. However, a pair of flux velocities (\({\mathbf{w}}_{p}\) and \({\mathbf{w}}_{t}\)) may be introduced as follows.

Fig. 19
figure 19

A diffusing magnetic field inside a sphere whose poloidal field lines are shrinking towards the toroidal line, while the toroidal field is diffusing towards the separator

Denoting poloidal and toroidal components by subscripts p and t, \({\mathbf{B}}_{p}\) and \({\mathbf{B}}_{t}\) change in time according to

$$\begin{aligned} \frac{\partial {\mathbf{B}}_{p}}{\partial t}=-{\varvec{\nabla }}\times {\mathbf{E}}_{t} \ \ \ \ \ \mathrm{and}\ \ \ \ \ \frac{\partial {\mathbf{B}}_{t}}{\partial t}=-{\varvec{\nabla }}\times {\mathbf{E}}_{p}. \end{aligned}$$

Thus, two separate dual flux velocities velocities \({\mathbf{w}}_{p}\) and \({\mathbf{w}}_{t}\) may be defined by

$$\begin{aligned} {\mathbf{E}}_{t}+{\mathbf{w}}_{p}\times {\mathbf{B}}_{p}={\mathbf{0}} \qquad {\mathrm {and}} \qquad {\mathbf{E}}_{p}+{\mathbf{w}}_{t}\times {\mathbf{B}}_{t}={\mathbf{0}}, \end{aligned}$$

which are perpendicular to \({\mathbf{B}}_{p}\) and \({\mathbf{B}}_{t}\), respectively. As it decays in time, the field in Fig. 19 behaves as if the poloidal field is shrinking at \({\mathbf{w}}_{p}\) towards the closed toroidal field line (C) and disappearing into the O-points of the poloidal field. At the same time the toroidal field can be regarded as shrinking and disappearing at the separator joining the null points \(N_{1}\) and \(N_{2}\).

4 The nature of reconnection in three dimensions

Before describing the different regimes for reconnection in 3D, we need to lay the groundwork by describing various fundamental concepts and the technique for classification. In addition, we enumerate the many ways in which reconnection in 3D is very different from reconnection in 2D.

Ohm’s law may be written in a non-ideal plasma as

$$\begin{aligned} {\mathbf{E}} + {\mathbf{v}} \times {\mathbf{B}} = {{\mathbf{N}}}, \end{aligned}$$

and it is the form of the nonideal term (\({\mathbf {N}}\)) that determines whether there is simple diffusion of magnetic field lines or 2D reconnection or 3D reconnection. In 2D the notion of a flux velocity is helpful in describing what happens, but in 3D it fails, although it is possible to replace it by a dual flux velocity (Sect. 3.3.3, Sect. 4.4).

4.1 Form of the non-ideal term for reconnection: diffusion and reconnection

First, suppose that the nonideal term \({\mathbf{N}}\) in Eq. (21) can be written as

$$\begin{aligned} {\mathbf{N}} = {\mathbf{u}} \times {\mathbf{B}} + {\varvec{\nabla }}{\varPhi }. \end{aligned}$$

Then the curl of Eq. (21) implies

$$\begin{aligned} \frac{\partial {\mathbf{B}}}{\partial t} = {{{\varvec{\nabla }}}} \times ({\mathbf{w}} \times {\mathbf{B}}), \end{aligned}$$

in terms of the flux velocity (\({{\mathbf{w}}}= {\mathbf{v}} - {\mathbf{u}}\)) and slippage velocity (\({\mathbf{u}}\)). In this case, the magnetic field may be said to move with the velocity \({\mathbf {w}}\). Thus we see that the nature of the evolution depends on the form of \({\mathbf{N}}\), as follows:

  1. (a)

    if \({\mathbf{N}} = {\mathbf{u}} \times {\mathbf{B}} + {\varvec{\nabla }}{\varPhi }\) and \({\mathbf{u}}\) is smooth, the magnetic field slips or diffuses through the plasma, but there is no reconnection;

  2. (b)

    if \({\mathbf{N}} = {\mathbf{u}} \times {\mathbf{B}} + {\varvec{\nabla }}{\varPhi }\) and \({\mathbf{u}}\) is singular at a point, then 2D reconnection takes place there;

  3. (c)

    if \({{\mathbf{N}}} \ne {\mathbf{u}}\times {\mathbf{B}} + {\varvec{\nabla }}{\varPhi }\), then reconnection occurs in 2.5D or 3D.

The relation between diffusion and reconnection is that reconnection implies diffusion (in an isolated region), but diffusion can take place without any reconnection.

4.2 Two-dimensional reconnection (\({\mathbf {E}}\cdot {\mathbf {B}}=0\))

Suppose that \({\mathbf {E}}\) is perpendicular to \({\mathbf {B}}\). Then from Eq. (21) \({\mathbf {N}}\) is also perpendicular to \({\mathbf {B}}\), and we have a two-dimensional situation. For instance, if \({\mathbf {v}}\) and \({\mathbf {B}}\) lie in the xy-plane and depend only on x and y, then \({\mathbf {E}}\) and \({\mathbf {N}}\) are in the z-direction. In this case we may write \({\varvec{\nabla }}{\varPhi } \equiv {\mathbf{0}}\), so that the slippage velocity is

$$\begin{aligned} {\mathbf{u}} = \frac{({\mathbf {B}}\times {\mathbf {N}})}{B^2}. \end{aligned}$$

Also, Eq. (21) reduces to

$$\begin{aligned} {\mathbf {E}}+ {\mathbf {w}}\times {\mathbf {B}}={\mathbf{0}}, \end{aligned}$$

and so there exists a flux velocity \({\mathbf {w}}_{\perp } = {\mathbf {E}}\times {\mathbf {B}}/B^{2}\) that conserves the flux and is smooth except possibly where the magnetic field vanishes. Three possibilities arise:

  1. (a)

    If \({\mathbf {B}}\ne {\mathbf{0}}\), then flux-conserving slippage of the magnetic field takes place, since \({\mathbf {w}}\) is smooth everywhere;

  2. (b)

    If there is an O-type null point at which \({\mathbf{E}}\ne 0\), then \({\mathbf {w}}\) possesses a divergent singularity at the O-point, where magnetic flux is destroyed or generated;

  3. (c)

    If there is an X-type null point at which \({\mathbf{E}}\ne 0\), then \({\mathbf {w}}\) possesses a hyperbolic singularity, where magnetic flux is reconnected and flux is weakly conserved with \({\mathbf {w}}\) regular except at the X-point.

4.3 Non-existence of a flux velocity in 3D

The notion of a flux velocity has been central to the theory of 2D reconnection, but it fails in 3D (when \({\mathbf {E}}\cdot {\mathbf {B}}\ne 0\)), since, for an isolated 3D nonideal region, a flux-conserving velocity (\({\mathbf {w}}\)) does not in general exist (Priest et al. 2003). The proof is straightforward. If a flux velocity does exist, then Eq. (22) holds and a function \({\varPhi }\) exists such that

$$\begin{aligned} {\mathbf {E}}+{\mathbf {w}}\times {\mathbf {B}}={\varvec{\nabla }}{\varPhi }. \end{aligned}$$

From this equation we may deduce that \({\mathbf {E}}\cdot {\mathbf {B}}={\varvec{\nabla }}{\varPhi } \cdot {\mathbf {B}}\), which may be integrated along a magnetic field line from one point (\({\mathbf {r}}_{1}\)) to another (\({\mathbf {r}}_{2}\)) on opposite sides of the diffusion region, to give

$$\begin{aligned} \int _{{\mathbf {r}}_{1}}^{{\mathbf {r}}_{2}}E_{\parallel }ds={\varPhi }({\mathbf {r}}_{2})-{\varPhi }({\mathbf {r}}_{1}). \end{aligned}$$

We may next assume that \(E_{\parallel }\) is, say, positive in the diffusion region, and so \({\varPhi }({\mathbf {r}}_{2})>{\varPhi }({\mathbf {r}}_{1})\). However, everywhere outside the isolated diffusion region, \({\mathbf {w}}={\mathbf {v}}\) and \({\varvec{\nabla }}{\varPhi }={\mathbf{0}}\), so that \({\varPhi }\) is uniform outside the diffusion region and therefore \({\varPhi }({\mathbf {r}}_{2})={\varPhi }({\mathbf {r}}_{1})\). Thus, we have a contradiction, which leads to the conclusion that a flux velocity does not exist, as required.

4.4 Fundamental differences between 2D and 3D reconnection

Magnetic reconnection in 3D is profoundly different from 2D with many new features. Among the aspects of reconnection in 2D that do not survive in 3D are the following:

  1. (2D:i)

    Reconnection occurs only at neutral points of X-type; these nulls may be pre-existing in the field and undergo local collapse to form a current sheet, or a one-dimensional current sheet may form, within which a 2D null is created when reconnection is initiated;

  2. (2D:ii)

    A flux velocity (\({\mathbf {w}}\)) exists and describes the speed at which magnetic flux moves, slipping relative to the plasma at a velocity \({\mathbf {w}}-{\mathbf {v}}\), but preserving the magnetic field line connections between points on the boundary; the exception is at an X-point, where \({\mathbf {w}}\) becomes singular and the connections change as the field lines break;

  3. (2D:iii)

    Before two flux tubes reconnect, they approach the diffusion region with velocity \({\mathbf {w}}={\mathbf {v}}\), and then they break and re-connect perfectly to create two new flux tubes, such that, once they have left the diffusion region, they move out at \({\mathbf {w}}={\mathbf {v}}\) (Fig. 20a);

  4. (2D:iv)

    When a magnetic flux tube has part of its length inside a diffusion region and part outside, the central section slips through the plasma with \({\mathbf {w}}\ne {\mathbf {v}}\), while the two wings of the flux tube are frozen to the plasma and move with \({\mathbf {w}}={\mathbf {v}}\) (Figs. 20b).

Fig. 20
figure 20

Image reproduced with permission from Priest et al. (2003), copyright by AGU

The nature of 2D reconnection. a Two flux tubes come in, break in the diffusion region (shaded) and re-connect perfectly. b While in the diffusion region, a flux tube slips through the plasma but preserves its connections, except at the X-point. For a related movie see Supplementary Information.

In contrast, 3D reconnection has the following properties:

  1. (3D:i)

    Magnetic reconnection may occur wherever sufficiently intense electric current concentrations form, for example at a null point, a separator or a quasi-separator (i.e., a hyperbolic flux tube) (Sects. 2.6, 5);

  2. (3D:ii)

    In general a single flux tube velocity (\({\mathbf {w}}\)) does not exist (Sect. 4.3), but it can be replaced by a pair (\({\mathbf {w}}_{\mathrm{in}},{\mathbf {w}}_{\mathrm{out}}\)) of flux velocities (called a dual flux velocity), which describe behaviour from two points of view, namely, a field line that enters a diffusion region or one that leaves it (Sect. 4.7.2);

  3. (3D:iii)

    In the absence of a null point or bald patch, the field-line mapping from one section of a boundary to another is continuous;

  4. (3D:iv)

    Magnetic field lines continually change their connections as part of them pass through a diffusion region; another way to view this is the following;

  5. (3D:v)

    In the process of reconnecting, two flux tubes split into four parts, each of which flips in a different manner (Fig. 21b); field lines that are projected through to the other side of a diffusion region flip, i.e., they move with a virtual velocity that differs from the plasma velocity (Sect. 4.3);

  6. (3D:vi)

    In general magnetic field lines do not break and reform pairwise, and neither do pairs of flux tubes (Figs. 21).

An illustrative example of this behaviour is presented in Sect. 4.7.2.

Fig. 21
figure 21

Image reproduced with permission from Priest et al. (2003), copyright by AGU. c Illustration of the principles in (a) and (b), wherein the evolution of two representative flux tubes from the solution of Hornig and Priest (2003) (see Sect. 4.7.2) are traced from ideal comoving footpoints (marked black); the dark sections of the tubes move at the local plasma velocity (outside D), while the light sections correspond to field lines that pass through the diffusion region. A localised diffusion region (shaded surface in the first frame) is present around the origin. From Pontin (2011). For a related movie see Supplementary Information

The nature of reconnection in 3D: a two magnetic flux tubes approach one another, break and partly rejoin, but b the projection of a flux tube beyond a diffusion region flips through the plasma in a virtual flow.

4.5 Classification and nature of 3D reconnection: general reconnection or singular reconnection

To reiterate the above, in 2D the basic features of magnetic reconnection are:

  1. (i)

    Two pairs of field lines approach an X-point, and are then broken and rejoined;

  2. (ii)

    There is an electric field \(({\mathbf{E}})\) perpendicular to the 2D plane;

  3. (iii)

    During reconnection, the magnetic connectivity of plasma elements is changed at the X-point surrounded by a localised non-ideal diffusion region.

In 3D, since reconnection can occur without null points or separatrices (contrary to (i)), Schindler et al. (1988) proposed a concept of “General Magnetic Reconnection” that is based on (iii) above and includes any process of local nonidealness that gives rise to an electric field component \((E_{\Vert })\) parallel to the magnetic field. Thus, the generalisation of (ii) above is that the integral

$$\begin{aligned} \int E_{\Vert } \ ds \ne 0, \end{aligned}$$

where the integral is performed along a magnetic field line. The maximum value of this integral over all field lines that thread the non-ideal region represents the rate of reconnection. Equivalently, 3D reconnection occurs if and only if the magnetic helicity (Sect. 4.6) changes in time. As shown by Schindler et al. (1988) and Hesse and Schindler (1988) when the condition (23) is met, plasma elements experience a change in magnetic field line connectivity.

With 3D reconnection requiring that the condition (23) be met, it is the formation of a localised current concentration that is the precondition for such reconnection. The reason why reconnection occurs at nulls, separators, quasi-separators (or hyperbolic flux tubes) and braids is that they are the natural locations where strong currents focus (see Sect. 5). It should be noted, however, that the nulls, separators and quasi-separators themselves are not the (only) locations for reconnection, since it occurs throughout the finite diffusion regions that surround them.

The theory of general reconnection has been exemplified, developed and applied in several directions. An analytical example of general magnetic reconnection has been presented by Schindler et al. (1988) that demonstrates how a plasmoid can be formed in 3D in the absence of a null point or separatrix. Furthermore, a mathematical formalism for General Magnetic Reconnection has been developed by Hesse and Schindler (1988), in which they express the magnetic field in terms of Euler potentials (\(\alpha ,\beta \)) and find equations for the time-behaviour of \(\alpha \) and \(\beta \).

Sub-classes of general magnetic reconnection may be defined in different ways, for example as driven and spontaneous reconnection, depending on whether motions of flux at large distances or local effects (such as enhanced resistivity or instability) dominate. Another way of categorising reconnection is in terms of the form of the nonideal term in Ohm’s law, which distinguishes slippage from 2D reconnection and 3D reconnection (See Sect. 4.1). While 2D reconnection occurs only at X-points, the properties of generic 3D reconnection (with a localised diffusion region within which \({\mathbf {E}}\cdot {\mathbf {B}}\ne 0\)) depend on whether a null-point is present (3D null reconnection) or not (3D non-null reconnection)—as discussed in more detail in Sects. 10-12.

General Reconnection includes examples of diffusion and slippage (such as in double layers or shock waves) that are not usually included in the concept of reconnection, and so it may be regarded as too general a concept. An alternative is therefore to restrict the definition of reconnection to Singular field-line Reconnection, in which the presence of \(E_{\parallel }\) along a field line is supplemented by the condition that the nearby field has a certain topology in a plane perpendicular to the field line (Priest and Forbes 1989; Hornig and Rastätter 1998). If the transverse topology is X-type, then we would refer to it as X-type Singular Reconnection, which is close in spirit to traditional 2D reconnection and includes most cases of separator and quasi-separator (or hyperbolic flux tube) reconnection. If the transverse topology is O-type, then we would refer to it as O-type Singular Reconnection, which includes Flux Tube Disconnection (Wilmot-Smith and Priest 2007) and some examples of separator reconnection discovered in numerical experiments (Parnell et al. 2010). For further details, see Priest and Forbes (2000).

4.6 Magnetic helicity and its changes during 3D reconnection

Magnetic helicity measures the twisting and kinking of a flux tube (called self-helicity), together with the linkage between different flux tubes (called mutual helicity). In an ideal medium, as a global topological invariant it does not change, whereas, in a resistive medium, it decays very slowly over the global magnetic diffusion time (\(\tau _{d}\)). During reconnection, magnetic helicity is approximately conserved, although it can be converted from one form to another: for example, mutual may be converted to self during the initial stages of a coronal mass ejection or eruptive solar flare, which may explain much of the twist that is observed (e.g., Priest and Longcope 2017). However, during 3D reconnection the tiny change in magnetic helicity that does take place is intimately related to its very occurrence, as demonstrated below.

Magnetic helicity is thought to be critical in the Sun’s cycle: continual photospheric motions energise the coronal magnetic field and increase the (modulus of the) helicity; and the coronal magnetic field evolves through states that are nearly force-free, building up magnetic helicity until it is ejected by coronal mass ejections (Heyvaerts and Priest 1984; Rust and Kumar 1994; Low 2001). An important step in this understanding was the development of a theory for relative helicity by Berger and Field (1984). Here we give definitions of magnetic helicity (Sect. 4.6.1, 4.6.2) and discuss its evolution (Sect. 4.6.3, 4.6.4).

4.6.1 Definition of magnetic helicity

For a closed volume (V) that is bounded by a surface S, with normal vector \({\mathbf{n}}\), the magnetic helicity is defined as

$$\begin{aligned} H_{0} = \int _{V} {{{\mathbf{A}} \cdot {\mathbf{B}}}}\; dV, \end{aligned}$$

where the magnetic field may be written \({{\mathbf{B}} = {\varvec{\nabla }}} \times {\mathbf{A}}\) in terms of the vector potential (\({\mathbf{A}}\)). Note that an arbitrary gauge \({\varvec{\nabla }} {\varPhi }_{A}\) may be added to \({\mathbf{A}}\) without changing \({\mathbf {B}}\), but it can be shown that the magnetic helicity is independent of this gauge, i.e., it is gauge invariant, provided that all magnetic field lines close within the volume (i.e., \({\mathbf{B}} \cdot {\mathbf{n}}=0\) on S).

However, if the boundary is open in the sense that magnetic fields enter or leave it, Berger and Field (1984) made an important breakthrough in realising that a relative magnetic helicity, defined as

$$\begin{aligned} H =\int _{V_{\infty }} {{{\mathbf{A}} \cdot {{\mathbf{B - A}}}}}_{0} \cdot {\mathbf{B}}_{0}\; dV, \end{aligned}$$

is also gauge invariant, where \({\mathbf{B}}_{0} = {{\varvec{\nabla }} \times {\mathbf{A}}}_{0}\) is a reference field with respect to which \(H_0\) is being measured, and \(V_{\infty }\) is the whole of space, including the volume both inside and outside V. \({\mathbf{B}}_{0}\) is usually taken to be potential inside V, to be identical with \({\mathbf{B}}\) outside V and to have \({{{\mathbf{A}} \times {\mathbf{n}} = {\mathbf{A}}}}_{0} \times {\mathbf{n}}\) on S.

4.6.2 Magnetic helicity of flux tubes

The magnetic helicity of a collection of flux tubes of magnetic flux \(F_{i}\) and twist \({\varPhi }_{T_{i}}\) may be written as a sum

$$\begin{aligned} H = \sum ^{N}_{i=1} H_{si} + \sum ^{N}_{{\mathop {i<j}\limits ^{i, j=1}}} H_{mij}, \end{aligned}$$

of self-helicities

$$\begin{aligned} H_{si} = \frac{{\varPhi }_{T_{i}} \ F_{i}^{2}}{2 \pi }, \end{aligned}$$

for each flux tube due to their own internal twist, and mutual helicities

$$\begin{aligned} H_{mij} = 2 L_{ij}\; F_{i} F_{j} , \end{aligned}$$

due to the linking of one tube with another, where \(L_{ij}\) is the linking number (Berger 1984).

For example, consider two coronal flux tubes having fluxes \(F_{A}\) and \(F_{B}\), with positive polarity footpoints \(A_{+}, B_{+}\) and negative footpoints \(A_{-}, B_{-}\) (Fig. 22). The mutual helicity may be written in terms of angles \(\angle B_{+} A_{-} B_{-}\) at footpoint \(A_{+}\) and \(\angle B_{+} A_{+} B_{-}\) at footpoint \(A_{+}\) as

$$\begin{aligned} H = (-\angle B_{+} A_{-} B_{-} + \angle B_{+} A_{+} B_{-}) \frac{F_{A} F_{B} }{2 \pi }. \end{aligned}$$

The structures in Fig. 22 then have helicities \((\theta _{1} - \theta _{2}) F_{A} F_{B} / \pi \) and \((\theta _{3} + \theta _{4}) F_{A} F_{B} / \pi \) in terms of the angles shown.

Fig. 22
figure 22

Nearby flux tubes have a mutual helicity of a \((\theta _{1} - \theta _{2} ) F_{A} F_{B} / \pi \), whereas crossing flux tubes have b \((\theta _{3} + \theta _{4} ) F_{A} F_{B} / \pi \) (after Berger 1999)

4.6.3 Rate of change of magnetic helicity

If the gauge of the vector potential is chosen such that \({\varvec{\nabla }\cdot {\mathbf {A}}}_{p} = 0\) and \({\mathbf{A}}_{p} \cdot {\mathbf{n}} = 0\) on the boundary S, the rate at which magnetic helicity changes may be written

$$\begin{aligned} \frac{dH}{dt} = -2 \int _{V} {{{\mathbf{E}} \cdot {\mathbf{B}}}}\; dV + 2 \int _{S} {\mathbf{A}}_{p} \times {{{\mathbf{E}} \cdot {\mathbf{n}}}}\; dS. \end{aligned}$$

Furthermore, when there is no diffusion on S and \({\mathbf{E}}\) is given by Ohm’s Law \(({{{\mathbf{E}} = - {\mathbf{v}} \times {\mathbf{B + j}}}} / \sigma )\), this reduces to

$$\begin{aligned} \frac{dH}{dt} = -2 \int _{V} {{\mathbf{j} \cdot {\mathbf{B}}}} / \sigma \; dV + 2 \int _{S} ({\mathbf{B} \cdot {\mathbf{A}}}_{p}) ({{{\mathbf{v}} \cdot \hat{\mathbf{n}}}) - ({\mathbf{v}} \cdot {\mathbf{A}}}_{p}) ({{{\mathbf{B}} \cdot \hat{\mathbf{n}}}}) \; dS, \end{aligned}$$

where \(\hat{\mathbf{n}}\) is the outwards normal to the volume.

The first term on the right of Eq. (28) represents resistive dissipation of magnetic helicity on a time-scale equal to the global diffusion time \((\tau _{d} = L^{2} / \eta )\), where L is the global length-scale over which the magnetic field varies. Thus, on times much smaller than this, if the boundary is closed so that the normal components of \({\mathbf {B}}\) and \({\mathbf {v}}\) vanish, the magnetic helicity is conserved. Furthermore, the second term represents the rate at which magnetic helicity is carried across the boundary by plasma motions normal to the boundary, whereas the third term shows that the magnetic helicity may be injected into the volume or extracted from it by lateral motions.

4.6.4 Magnetic helicity changes during 3D reconnection

Since reconnection involves magnetic diffusion only in a diffusion region (\(D_R\)), which is a small part of the volume, the total magnetic helicity is approximately conserved during reconnection. However, the very occurrence of three-dimensional reconnection is directly related to the small change in magnetic helicity that takes place. Consider Eq. (27). Since the electric field is perpendicular to the magnetic field (\({{{\mathbf{E}} \cdot {\mathbf{B}}}} = 0\)) everywhere except in the diffusion region \(D_{R}\), the first term on the right reduces to an integral over \(D_{R}\) alone. If also the electric field vanishes (\({\mathbf{E}} = 0\)) on the surface S, the second term vanishes, and so the equation becomes

$$\begin{aligned} \frac{d H }{ dt} =-2 \int _{D_{R}} {{{\mathbf{E}} \cdot {\mathbf{B}}}}\; dV=-2 \int _{D_{R}} E_{\parallel }B\; dV. \end{aligned}$$

Recall that, by definition, 3D reconnection takes place when the parallel electric field is non-zero (\( E_{\parallel } \ne 0\)) in \(D_{R}\), and so from Eq. (29) the occurrence of such reconnection is equivalent to there being a change of magnetic helicity H with time. Furthermore, if \(V_{R}\) is the diffusion region volume and \(\delta H\) is the change of magnetic helicity in a time \(\delta t\), the parallel electric field is

$$\begin{aligned} E_{\parallel } \simeq \frac{1}{2V_{R} B_{0}} \frac{\delta H}{\delta t}. \end{aligned}$$

The helicity is then approximately conserved even in an evolution involving 3D reconnection because at high \(R_m\) the current layers are very thin, so that \(V_R\) is very small compared to the domain volume.

Although the total helicity in a volume is approximately conserved during reconnection, it may be redistributed within a configuration. When the field is modelled as being composed of discrete flux tubes this can be interpreted as a change from mutual helicity to self helicity or vice versa. Alternatively, for continuous fields the notion of a field line helicity (Antiochos 1987; Berger 1988) can be useful in interpreting how the linkage of flux is changed during reconnection. The field line helicity is defined as

$$\begin{aligned} {\mathcal {A}}=\int _{F({\mathbf {x}})}{\mathbf {A}}\cdot d{\mathbf{l}}, \end{aligned}$$

where \(d{\mathbf{l}}\) is tangent to the field line \(F({\mathbf {x}})\). For a suitable choice of gauge, it measures the net linkage of all field lines in the domain with the field line of interest, while a flux-weighted integral of \({\mathcal {A}}\) over all field lines yields the helicity of the volume, justifying the name (Berger 1988; Russell et al. 2015). Moreover, its distribution uniquely describes the magnetic topology of the field (Yeates and Hornig 2013). It has been used, for example, to characterise the structure and evolution of active regions (Moraitis et al. 2019, 2021). Russell et al. (2015) investigated the way in which the distribution of field line helicity can be changed by reconnection within the volume, although this remains to be fully explored.

4.7 Techniques for modelling 3D reconnection

In three dimensions, reconnection may take place in a variety of different regimes and a variety of different geometries, namely, at a null point (Sect. 10), at a separator (Sect. 11) or at non-null regions such as a quasi-separator (Sect. 12) or a braided field (Sect. 14.1). Which of these regimes occurs depends both on the geometry (i.e., whether the magnetic configuration contains a null point, a separator, a quasi-separator or a braid) and on the nature of the plasma flow. The properties of these different regimes are being discovered by a combination of several complementary techniques that we describe here, namely, kinematic models that are either ideal (Sect. 4.7.1) or resistive (Sect. 4.7.2) and numerical computations (Sect. 4.7.3).

4.7.1 Kinematic ideal models

The full MHD equations for determining the plasma velocity (\({\mathbf {v}}\)) and magnetic field (\({\mathbf {B}}\)) include the induction equation and the equation of motion, but in a kinematic approach one neglects the equation of motion and considers the implications of just the induction equation by either imposing \({\mathbf {v}}\) and determining \({\mathbf {B}}\) or imposing \({\mathbf {B}}\) and solving for \({\mathbf {v}}\). Lau and Finn (1990) and Priest and Titov (1996) began by exploring the nature of the flow in the ideal region around a diffusion region during steady 3D reconnection. They prescribed the form of the magnetic field (\({\mathbf {B}}\)) and solved the kinematic equations for the plasma velocity (\({\mathbf {v}}\)) and electric field (\({\mathbf {E}}\)), namely,

$$\begin{aligned} {\mathbf {E}}+{\mathbf {v}}\times {\mathbf {B}}={\mathbf{0}} \ \ \ \mathrm{(a)}\ \ \ \ {{\mathrm {and}}} \ \ \ {{{\varvec{\nabla }}}} \times {\mathbf {E}}= {\mathbf{0}}.\ \ \ \mathrm{(b)} \end{aligned}$$

First of all, Eq. (31b) implies that \({\mathbf {E}}={\varvec{\nabla }}{\varPhi }\) and so the component of Eq. (31a) perpendicular to \({\mathbf {B}}\) yields

$$\begin{aligned} {\mathbf {B}}\cdot {\varvec{\nabla }}{\varPhi }=0, \end{aligned}$$

which implies that \({\varPhi }\) is constant along magnetic field lines. If the values of \({\varPhi }\) are imposed on a surface (S) surrounding the ideal region, then Eq. (32) may be integrated along field lines (characteristics) to determine the value of \({\varPhi }\) (and therefore \({\mathbf {E}}\)) throughout the volume. A variety of different boundary conditions may be imposed on the surface S, and so give rise to different solutions. Next, the component of Eq. (31a) perpendicular to \({\mathbf {B}}\) determines the plasma velocity normal to \({\mathbf {B}}\) everywhere as

$$\begin{aligned} {\mathbf {v}}_{\perp }=\frac{{\varvec{\nabla }}{\varPhi } \times {\mathbf {B}}}{B^{2}}. \end{aligned}$$

This approach was used to show that current is likely to become focused along either the spine or the fan of a null point or along a separator and so to lead to different null-point reconnection regimes, as described in Sects. 10.1.1, 11.

4.7.2 Kinematic resistive modelling

However, the next step is to model what goes on inside a diffusion region. Hornig and Priest (2003) developed a formalism for modelling kinematically an isolated 3D diffusion region in which flux freezing breaks down, and they applied their formalism to a case without null points, although it was later applied to null points (Sect. 10.1.2) and separators. They solved

$$\begin{aligned} {\mathbf {E}}+{\mathbf {v}}\times {\mathbf {B}}=\eta {\mathbf {j}}, \end{aligned}$$

where \({{{\varvec{\nabla }}}} \times {\mathbf {E}}= {\mathbf{0}}\), \({\mathbf {j}}={{{\varvec{\nabla }}}} \times {\mathbf {B}}/\mu \) and \({{{\varvec{\nabla }}}} \cdot {\mathbf {B}}=0\). Their idea was to impose a sufficiently simple magnetic field that both the mapping and the inverse mapping of the magnetic field can be found analytically. Then, after writing \({\mathbf {E}}={\varvec{\nabla }}{\varPhi }\), the integral of the component of Eq. (34) parallel to \({\mathbf {B}}\) determines \({\varPhi }\) everywhere as

$$\begin{aligned} {\varPhi } =\int \frac{\eta \ {\mathbf {j}}\cdot {\mathbf {B}}}{B}\ ds +{\varPhi }_{e} = \int \eta \ {\mathbf {j}}\cdot {\mathbf {B}}\ ds^* +{\varPhi }_{e}, \end{aligned}$$

where \({\varPhi }_{e}\) is the imposed value of \({\varPhi }\) at one end of the field lines, s is the distance along field lines, and \(s^*\) is a stretched distance such that \(ds^*=ds/B\). Then the flow normal to the field lines is determined by the component of Eq. (34) perpendicular to \({\mathbf {B}}\) as

$$\begin{aligned} {\mathbf {v}}_{\perp }=\frac{({\varvec{\nabla }}{\varPhi } -\eta \ {\mathbf {j}}) \times {\mathbf {B}}}{B^{2}}. \end{aligned}$$

Here we illustrate this approach by describing the solution of Hornig and Priest (2003), who considered the simple QSL field

$$\begin{aligned} {\mathbf {B}}= \frac{B_0}{L}\left( y\ {{\hat{\mathbf{x}}}}+k^2 x\ {{\hat{\mathbf{y}}}} + {{\hat{\mathbf{z}}}}\right) , \end{aligned}$$

namely, a 2D X-point field in the xy-plane with a uniform field in the z-direction, and a uniform current \({\mathbf {j}}= (k^2-1)B_0/L \ {{\hat{\mathbf{z}}}}\). The equations (\({\mathbf{X}} ({\mathbf{x}}_0, s^*)\)) of field lines whose initial point is \({\mathbf{x}}_0\) follow from integrating \(d{\mathbf{X}}/ds^*={\mathbf {B}}\) as

$$\begin{aligned} X= & {} x_0 \cosh (B_0ks^*/L)+(y_0/k)\sinh (B_0ks^*/L), \nonumber \\ Y= & {} y_0 \cosh (B_0ks^*/L)+x_0 k\sinh (B_0ks^*/L), \nonumber \\ Z= & {} z_0+B_0s^*, \end{aligned}$$

while the inverse mapping is given by

$$\begin{aligned} X_0= & {} x \cosh (B_0ks^*/L)-(y/k)\sinh (B_0ks^*/L), \nonumber \\ Y_0= & {} y \cosh (B_0ks^*/L)-x k\sinh (B_0ks^*/L), \nonumber \\ Z_0= & {} z-B_0s^*. \end{aligned}$$

In order to obtain a localised non-ideal term (\(\eta {\mathbf {j}}\)), since \({\mathbf {j}}\) is constant, it is necessary to localise the resistivity, and in order to obtain an analytical solution for \({\varPhi }\), they chose the following form

$$\begin{aligned} \eta (x_0,y_0,s^*) = \eta _0 \exp \left( -\frac{B_0^2{s^*}^2+x_0^2+y_0^2}{l^2}\right) , \end{aligned}$$

where \(z_0=0\), so that \(x_0,y_0\) are coordinates of field lines in the plane \(z=0\). Thus, \(\eta \) is positive with a maximum value of \(\eta _0\) at the origin, and it decreases exponentially with distance from the origin. The diffusion region D has the shape of a sphere distorted towards a tetrahedron (Fig. 23a).

Fig. 23
figure 23

Image reproduced with permission from Hornig and Priest (2003), copyright by AIP

a The diffusion region D (shaded) at 2% of the maximum diffusivity, and the hyperbolic flux tube that encloses it when \(k=2,\ L=10,\ l=1\). b In the \(z=0\)-plane, the field lines and the difference (\({{\mathbf {w}}}^{\mathrm{in}}-{{\mathbf {w}}}^{\mathrm{out}}\)) between the flux velocities of field lines anchored above and below D.

Then Eq. (35) may be integrated to give

$$\begin{aligned} {\varPhi }(x_0,y_0,s^*)=\frac{\sqrt{\pi }B_0\ \eta _0\ l\ (k^2-1)\ \mathrm{erf} (B_0s^*/l)}{2L \exp ((x_0^2+y_0^2)/l^2)} + {\varPhi }_e(x_0,y_0), \end{aligned}$$

where \((x_0,y_0,s^*)\) is given in terms of (xyz) by Eq. (39) and \({\mathbf {v}}_{\perp }\) follows from Eq. (36). Furthermore, a component of velocity parallel to \({\mathbf {B}}\) can be added to make \(v_z=0\), say. The freedom of being able to choose \({\varPhi }_e(x_0,y_0)\) is linked with the following splitting of Ohm’s law into non-ideal and ideal parts:

$$\begin{aligned} {\varvec{\nabla }}{\varPhi }_{\mathrm{nid}}+{\mathbf {v}}_{\mathrm{nid}} \times {\mathbf {B}}= & {} \eta {\mathbf {j}}, \end{aligned}$$
$$\begin{aligned} {\varvec{\nabla }}{\varPhi }_{\mathrm{id}}+{\mathbf {v}}_{\mathrm{id}} \times {\mathbf {B}}= & {} {\mathbf{0}}. \end{aligned}$$

Depending on the choice of \({\varPhi }_e(x_0,y_0)\), we can consider pure solutions or composite solutions.

For a pure solution, set \({\varPhi }_e(x_0,y_0)\equiv 0\), so that \({\mathbf {v}}\) vanishes when \(z=0\). In this case, counter-rotating flows are produced in the ideal regions above and below the plane \(z=0\) (Fig. 23b). Such flows are confined to a region, a hyperbolic flux tube (HFT) (Sect. 2.6.2), that contains only the field lines that thread the diffusion region D, as indicated in Fig. 23a. The flows are circular close to \(z=0\), but become highly elongated at large distances from D. They are associated with flipping and with the small change in magnetic helicity essential to 3D reconnection (Sect. 4.6.3).

The rate of reconnection of flux is calculated by evaluating the integral

$$\begin{aligned} \frac{d{\varPhi }_{\mathrm{mag}}}{dt} =\int E_{\parallel } ds \end{aligned}$$

along a field line through the diffusion region (Schindler et al. 1991; Hesse et al. 2005). For this example this turns out to be \(2{\varPhi }(0,0,\infty )\), which has quite a different interpretation from the normal 2D picture, where flux is cut and reconnected at one location in the diffusion region (D), namely, an X-point, and there is a unique flux velocity. Instead, in 3D, every field line in the HFT continually changes its connection in D, and two flux-conserving velocities (\({\mathbf {w}}^{\mathrm{in}}\) and \({\mathbf {w}}^{\mathrm{out}}\)) are needed, one of which describes the motion of field lines attached to plasma elements above D, while the other represents the velocity of field lines attached to plasma below it (Sect. 4.4). Thus, for a given \({\mathbf {B}}\), \({\mathbf {w}}^{\mathrm{in}}\) satisfies

$$\begin{aligned} {\mathbf {E}}^{\mathrm{in}}+{\mathbf {w}}^{\mathrm{in}}\times {{\mathbf {B}}={\mathbf{0}}}, \ \ \mathrm{where}\ \ {{\mathbf {E}}}^{\mathrm{in}}={\varvec{\nabla }}{\varPhi }^{\mathrm{in}}, \ \ \mathrm{so\ that} \ \ {{\mathbf {w}}}^{\mathrm{in}}_{\perp }=\frac{{\varvec{\nabla }}{\varPhi }^{\mathrm{in}}\times {\mathbf {B}}}{B^2}, \end{aligned}$$

while \({\mathbf {w}}^{\mathrm{out}}\) satisfies

$$\begin{aligned} {\mathbf {E}}^{\mathrm{out}}+{\mathbf {w}}^{\mathrm{out}}\times {{\mathbf {B}}={\mathbf{0}}}, \quad \text {where} \quad {{\mathbf {E}}}^{\mathrm{out}}={\varvec{\nabla }}{\varPhi }^{\mathrm{out}}, \quad \text {so that} \quad {{\mathbf {w}}}^{\mathrm{out}}_{\perp }=\frac{{\varvec{\nabla }}{\varPhi }^{\mathrm{out}}\times {\mathbf {B}}}{B^2}. \end{aligned}$$

Outside D, \({\mathbf {w}}^{\mathrm{in}}\) is just the same as \({\mathbf {v}}\) on one side of D, while \({\mathbf {w}}^{\mathrm{out}}\) is the same on the other side, so for \({\varPhi }^{\mathrm{in}}\) we can just choose the asymptotic value of \({\varPhi }\) from Eq. (41) on one side, say, \({\varPhi }(x_0,y_0,-\infty )\), and for \({\varPhi }^{\mathrm{out}}\) we choose the asymptotic value on the other side, namely, \({\varPhi }(x_0,y_0,\infty )\) or

$$\begin{aligned} {\varPhi }^{\mathrm{in}} = -\frac{\sqrt{\pi }B_0 \eta _0l(k^2-1)}{2L \exp ((x_0^2+y_0^2)/l^2)}, \qquad {\varPhi }^{\mathrm{out}} = \frac{\sqrt{\pi }B_0 \eta _0l(k^2-1)}{2L \exp ((x_0^2+y_0^2)/l^2)}. \end{aligned}$$

The rate of mismatching of flux in, say, the \(z=0\) plane is then the difference between \({\mathbf {w}}^{\mathrm{in}}\) and \({\mathbf {w}}^{\mathrm{out}}\) there, namely,

$$\begin{aligned} {\varDelta } {{\mathbf {w}}}_{\perp }= {{\mathbf {w}}}^{\mathrm{out}}_{\perp }-{{\mathbf {w}}}^{\mathrm{in}}_{\perp }=\frac{({\varvec{\nabla }}{\varPhi }^{\mathrm{out}}-{\varvec{\nabla }}{\varPhi }^{\mathrm{in}})\times {\mathbf {B}}}{B^2}= 2\frac{{\varvec{\nabla }}{\varPhi }^{\mathrm{out}}\times {\mathbf {B}}}{B^2}. \end{aligned}$$

The corresponding vector field is shown in Fig. 23b. The resulting rate at which magnetic flux crosses a radial line between the origin and the boundary of D is just the potential difference across the line, namely,

$$\begin{aligned} {\varDelta } {\varPhi }_{{\varDelta } w} = 2{\varPhi }^{\mathrm{out}} = 2{\varPhi }(0,0,\infty ), \end{aligned}$$

which is identical to the reconnection rate and so provides us with a physical interpretation.

For a composite solution, any ideal flow may be added to the pure flow, and particularly useful is a stagnation flow,

$$\begin{aligned} {\varPhi }_{\mathrm{id}}=-{\varPhi }_0\frac{x_0\ y_0}{l^2}, \end{aligned}$$

since it carries magnetic flux into the diffusion region (D), lets it reconnect, and then removes it from D. The field line behaviour is then governed by a combination of the external stagnation flow and the inherent counter-rotational flow associated with the “pure” reconnection solution. As described by Hornig and Priest (2003), different field line behaviours are possible in different regimes. Field lines transported into the non-ideal region tend to split in two (when followed from two plasma elements above and below the diffusion region), before exiting the diffusion region connected to different partners, as illustrated in Figure 21(b,c).

These solutions may be regarded as either kinematic (i.e., satisfying just the induction equation) or as fully dynamic in the limit of uniform density and slow flow (since they also satisfy the equations \({{{\varvec{\nabla }}}} \cdot {\mathbf {v}}=0\) and \({\varvec{\nabla }}p={\mathbf {j}}\times {\mathbf {B}}\)). This resistive kinematic approach has also been used to discover different reconnection regimes near null points, as described in Sect. 10.1.2.

4.7.3 Computational modelling

Although the analytical approach of kinematic modelling has been invaluable in pointing the way and suggesting what kinds of 3D reconnection are likely, computational experiments have been crucial in going beyond the limitations of the kinematic approach by solving the full set of MHD equations and revealing many new features. These experiments build on previous examples of 2D and 2.5D modelling.

The phrase “2D reconnection” refers to a strictly two-dimensional field [\(B_x(x,y), B_y(x,y)\)] that varies in two dimensions, whereas 2.5D has a field of the form [\(B_x(x,y), B_y(x,y), B_z(x,y)\)] with a guide field [\(B_z(x,y)\)], and “3D reconnection” refers to a fully 3D field [\(B_x(x,y,z), B_y(x,y,z), B_z(x,y,z)\)]. Thus, a 2D field should not be confused with a 2.5D field, which is topologically unstable in the sense that a small general 3D perturbation will destroy its topological structure. Many useful theories and simulations in 2D and 2.5D have helped clarify our understanding of reconnection, but most examples in nature are three-dimensional, and so the 2D and 2.5D understanding is likely to be only partial. For example, when a 2.5D simulation models so-called “anti-parallel” reconnection, it is likely to represent in reality a local snapshot of what in 3D would be null-point reconnection, whereas so-called “component” reconnection should more properly be referred to as a local snapshot of separator or quasi-separator reconnection. Numerical experiments have shown that the current \({\mathbf{j}}\) naturally builds up at null points, separators, quasi-separators and in braids—see Sect. 5—and so naturally leads to reconnection at these locations. What’s more, dedicated simulations at each of these structures has helped to reveal the rich behaviour of these different 3D reconnection regimes. These simulations and reconnection regimes—as well as applications to understanding observations from the Sun and beyond—are discussed in detail in Sects. 1015.

5 Formation of current sheets

Magnetic reconnection may occur in 3D fields wherever sufficiently intense current concentrations form (property (3D:i) in Sect. 4.4). Thus, to understand where reconnection takes place we must first determine where and how currents accumulate. Typically, the current accumulates in narrow layers called current sheets, across which there is a change in the tangential component of the magnetic field. In this section, we treat such sheets as discontinuities. In practice, however, they are resolved by diffusion processes and so possess a small finite width, which is modelled in the sections that follow on magnetic annihilation and reconnection. It should also be noted that in this section we focus on slow formation of current sheets through quasi-static states, but they may also form by a dynamic local “collapse” process as described by Forbes et al. (1982) (Sect. 6.3).

When a perturbed magnetic configuration evolves towards an equilibrium with partial or no reconnection (due to time constraints or microscopic limitations), the new equilibrium will often contain current sheets. Such sheets may later dissipate as reconnection transfers magnetic flux from one topological region to another (Sect. 7) or they may go unstable to tearing mode instability (Sect. 8). Here we suppose for simplicity that no reconnection takes place and describe techniques to model current sheet appearance. First of all, we focus on 2D current sheets, where an elegant and powerful technique was discovered by Green (1965), in which the sheets are treated as cuts in a complex plane when the surrounding field is potential. Then we discuss what happens in 2.5D and 3D configurations.

5.1 Current sheets in 2D potential fields

Here we describe static current sheets in 2D potential fields. The results can be applied to quasi-static formation or the evolution of such sheets through a series of equilibria. They can also be modified to allow for reconnection of a given amount of flux.

In 2D, a current-free (i.e., potential) magnetic field obeys

$$\begin{aligned} \frac{\partial B_{y}}{\partial x} - \frac{\partial B_{x} }{\partial y} = 0 \qquad \text {and} \qquad \frac{\partial B_{x} }{\partial x} + \frac{\partial B_{y} }{ \partial y} = 0, \end{aligned}$$

which are satisfied by the following combination of its components \((B_x(x,y)\), \(B_y(x,y))\)

$$\begin{aligned} B_{y} + iB_{x} = f(z), \end{aligned}$$

where \(z=x+iy\) is the complex variable and f(z) is any analytic function of z. Thus, for example, a linear X-point field \(B_{x}=y, B_{y}=x\) (Fig. 24a) may be written as

$$\begin{aligned} B_{y} + iB_{x} =x+iy \equiv z. \end{aligned}$$

Then the question arises: if a motion of the sources of the magnetic field leads to the formation of a series of equilibria with a current sheet growing from the X-point and containing Y-points at its ends (\(z=\pm iL\)), as in Fig. 24b, what is the best way to describe the equilibria? Green (1965) came up with an elegant answer, namely, to write the field as

$$\begin{aligned} B_{y} + iB_{x} = (z^{2} + L^{2}) ^{1/2}, \end{aligned}$$

with a cut in the complex plane from one end of the current sheet to the other. This behaves like z at large distances, and the separatrix field lines through the Y-points are inclined at an angle \(2 \pi / 3\).

Fig. 24
figure 24

The magnetic field a near an X-type null point which evolves to a field with a current sheet having at its ends either b Y-points or c reversed currents and singularities

A more general solution with singularities at the ends of the sheet when \(a\ne L\) was discovered by Somov and Syrovatsky (1976), namely,

$$\begin{aligned} B_{y} + iB_{x} = \frac{z^{2} + a^{2} }{ (z^{2} + L^{2})^{1/2}}, \end{aligned}$$

where \(a^{2} < L^{2}\) (Fig. 24c), and was later generalised further by Bungey and Priest (1995).

Other extensions have generalised the above 2D models in a variety of ways. Some of these are summarised briefly in the following, and in much more detail by, e.g., Priest and Forbes (2000). First, a two-dipole field geometry external to the sheet has been considered as a model for two bipolar regions on the Sun with either a planar or curved current sheet (e.g., Sweet 1958a; Priest and Raadu 1975). Such current sheets have been invoked in models of coronal loops (Low 1981, 1986) and solar prominences (e.g., Kippenhahn and Schluter 1957; Malherbe and Priest 1983). Priest et al. (1995) included time dependence in the model describing the nonlinear evolution for the dynamic time-dependent formation of a current sheet by solving the low-beta equation of motion and the ideal induction equation. Titov (1992) developed a clever technique for computing 2D potential fields with multiple current sheets by placing image sheets below the photosphere.

Furthermore, Priest and Syntelis (2021) have described a technique for dealing with 2D sheets without resorting to complex variable theory, which is invaluable because it may then be generalised to 3D. The approach involves constructing a current sheet from an infinite set of line currents. This method is illustrated in the following section for modelling an axisymmetric field at a separator ring. Finally, models for 2D current sheets may be extended to the case of finite plasma pressure. In this case the field will relax towards a force-free state in which the total pressure \((p+B^2/(2\mu ))\) balances across the separatrices, leading to different geometries such as cusp-shaped and curved separatrices (e.g., Bajer 1990; Bungey and Priest 1995).

5.2 Lateral or shearing motion of 2.5D fields to form sheets at separators and separatrices

When going beyond 2D potential fields to model magnetic fields that are 3D potential or force-free or magnetohydrostatic, current sheets can no longer be treated with complex variable theory and so other approaches are needed. By a “2.5D” field we mean a field such as (\(B_x(x,y), B_y(x,y), B_z(x,y)\)) with all three magnetic components that vary with only two variables (xy).

5.2.1 Three-dimensional axisymmetry near a separator

The magnetic field of a 3D axisymmetric annular current sheet created between two approaching dipoles was first analysed by Tur (1977) and Longcope and Cowley (1996). Indeed, Longcope and Cowley (1996) considered the topological admissibility of sequences of equilibria (as done for 2D X-points by Syrovatsky 1971), and argued that equilibria containing tangential discontinuities at the separator should result from certain perturbations. Building on their ideas, Priest and Syntelis (2021) developed a model in which the magnetic field is written in terms of cylindrical polar coordinates \((R,\phi ,z)\), as

$$\begin{aligned} (B_R,B_z)=(z,r)+(b_{SR}(R,z),b_{Sz}(R,z)), \end{aligned}$$

due to a ring of X-points near \(R=R_0\) and the field \((b_{SR},b_{Sz})\) of the current sheet itself. The current \(J\hat{\varvec{\phi }}\) in the sheet is related to the R-component (\(B_S\)) of the field at the edge of the sheet by

$$\begin{aligned} \mu J(R)=2B_S(R). \end{aligned}$$

The profile of \(B_S\) and therefore of current J that makes the normal component (\(B_z(R_0^{\prime })\)) of magnetic field vanish at the current sheet is then given by

$$\begin{aligned} R=-\lim _{{z} \rightarrow 0} b_{Sz}(R,z) \end{aligned}$$

at the sheet, which is found by first calculating the field due to a current ring and then summing over an infinite set of infinitesimal current rings to find the field of the current sheet.

To lowest order in \((r^2+z^2)^{1/2}/R_0^{\prime }\), the z-component of the magnetic field close to a current ring is

$$\begin{aligned} B_z \approx -\frac{\mu I_0}{4\pi }\left\{ \frac{2r}{r^2+z^2}+\frac{1}{R_0^{\prime }}\log _e\frac{(z^2+r^2)^{1/2}}{8R_0^{\prime }}\right\} , \end{aligned}$$

where \(r=R-R_0^{\prime }\). The first term is the field due to a straight current, while the second term gives the effect of the curvature of the current ring, which decreases the field on the outside of the ring and increases it on the inside of the ring, as shown in Fig. 25(a). After summing over a series of infinitesimal current rings to give a current sheet, the condition (Eq. 49) for the tangential field to vanish at the sheet becomes

$$\begin{aligned} R=\frac{1}{\pi }\lim _{{z} \rightarrow 0} \int _{-L/2}^{L/2}{\textstyle {\frac{1}{2}}B_S(r_0^{\prime })} \left\{ \frac{2(r-r_0^{\prime })}{z^2+(r_0^{\prime }-r)^2}+\frac{1}{R_0}\log _e\frac{[z^2+(r_0^{\prime }-r)^2)]^{1/2}}{8R_0}\right\} dr_0^{\prime }. \end{aligned}$$

This integral equation has been solved by Priest and Syntelis (2021) to give the profile of \(B_S(R)\) and applied to the problem of chromospheric and coronal heating by photospheric flux cancellation. Longcope (1996) used the formation of current sheets along separators as the basis for a model for reconnection, flaring and heating in the corona.

Fig. 25
figure 25

Image reproduced with permission from Priest and Syntelis (2021), copyright by ESO

a The magnetic field near part of a circular current loop of current \(I_0\). bd The magnetic field near an axisymmetric current sheet due to the sum of (b) a ring of nulls at radius \(R_0\) and c a current sheet of length L or equivalently d a set of current rings.

5.2.2 Shearing of separatrices of a 2.5D field

Current sheets may also be created when a separatrix touches a boundary at a bald patch (Fig. 26a) (Titov et al. 1993). Converging motions will give rise to a current sheet extending upwards, as in Fig. 26b, but shearing motions produce a long curved sheet stretching all along the separatrix (Fig. 26c) (Low and Wolfson 1988; Vekstein et al. 1991; Amari and Aly 1990). This may be analysed by considering a 2.5D equilibrium field of the form \((B_{x}, B_{y}, B_{z}) = (\partial A / \partial y, - \partial A / \partial x, B_{z}(A))\), where the force-free condition \({{{\mathbf{j}} \times {\mathbf{B}}}} = {\mathbf{0}}\) gives rise to the Grad–Shafranov equation for the flux function (A), namely,

$$\begin{aligned} \nabla ^{2} A+B_{z} \frac{dB_{z} }{dA} = 0. \end{aligned}$$

If a smooth footpoint displacement of a potential field of the form \({\mathbf{B}}_{p} = (\partial A / \partial y, - \partial A / \partial x)\) is imposed, the integral of the field-line equation becomes \( B_{z} (A) = {d(A) }/{V(A)}, \) in terms of the difference d(A) in footpoint displacement out of the plane and the differential flux volume \(V(A) = \int ds / B_{p}\). Although \(B_{z}(A)\) is constant along a given field line, its values on field lines above and below the separatrix AOB may be quite different, which turns the whole separatrix into a current sheet (Fig. 26c). Similarly, a 2D field with an X-point (Fig. 26d) is transformed by shearing motions into two cusp points at the ends of a current sheet extending all along the separatrices (Vekstein and Priest 1992, 1993) (Fig. 26e).

Fig. 26
figure 26

Image reproduced with permission from Priest (2014), copyright by CUP

A simple current sheet (thick curve) is produced in (b) by the effect of converging motions on (a) a quadrupolar field with no X-point, but the effect of (c) shearing motions is to create a current sheet all along the separatrix. Similarly, if there is an X-point present, shearing motions also create (e) a separatrix current sheet.

5.3 Magnetic relaxation

The preceding sections describe analytical models for current sheets in magnetic fields with certain symmetries. A useful technique for producing magnetic equilibria in more general geometries—often containing current sheets—is “magnetic relaxation” (e.g., Sturrock and Woodbury 1967; Arnol’d 1974; Klimchuk et al. 1988). Conceptually, the idea is to begin with a magnetic configuration that is not in equilibrium (and does not contain current sheets) and then to lower the total energy and so move towards an equilibrium while maintaining the magnetic field line topology (i.e., prohibiting reconnection). For example, Moffatt (1985) described the formation of current sheets in a three-dimensional configuration consisting of two linked flux tubes.

Computational implementations of ideal magnetic relaxation take advantage of the fact that the magnetic field evolves like a line element in a flow and use a Lagrangian computational mesh. In this way, the magnetic topology can be exactly maintained, since the expression for \({\mathbf {B}}\) is derived directly from the numerical mesh deformation (Craig and Sneyd 1986). Various different evolution equations can be used (Candelaresi et al. 2015), but, since it is the final state that is of interest rather than the evolution towards it, evolutions that minimise the magnetic energy in the most efficient way are chosen in practice. A common choice is

$$\begin{aligned} \rho \frac{\partial {\mathbf{v}} }{ \partial t} = - {{\varvec{\nabla }}} p + {{\mathbf{j}} \times {\mathbf{B}}} - K {\mathbf{v}}, \end{aligned}$$

where K is a friction coefficient, with the relaxation being described as magnetofrictional (Chodura and Schlueter 1981; Craig and Sneyd 1986). This choice of evolution equation has the great advantage that the total magnetic energy decreases monotonically in time. Alternatively, the energy may be damped by a viscous term, in which case the sum of the magnetic and kinetic energies decays monotonically (Moffatt 1985). In either case, the magnetic energy tends to a finite limit, which is non-zero if the initial topology is nontrivial in the sense that not all of the field lines can shrink to a point without cutting other field lines (see, e.g., Fig. 27).

Fig. 27
figure 27

Image reproduced with permission from Pontin and Hornig (2020), copyright by the authors, based on the simulations of Wilmot-Smith et al. (2009b)

Illustration of the Lagrangian relaxation computational approach, showing some representative grid lines (grey) and magnetic field lines both prior to relaxation (left) and in the numerically obtained equilibrium (right). The grid deforms to allow field lines to equilibrate the twist or stress along their length.

The final state of such a relaxation (with the lowest limit for the magnetic energy) is determined by the magnetic field topology. Depending on the evolution equation used, the final state may be force-free (\({\mathbf{j}} \times {\mathbf{B}} ={\mathbf{0}}\)) or magnetostatic (\({\mathbf{j}} \times {\mathbf {B}}= {\varvec{\nabla }} p\)) and may possess current sheets, across which \({{{\mathbf{n}} \cdot {\mathbf{B}}}} = 0\) and the total pressure \((p + B^{2} / (2 \mu ))\) is continuous. This technique has been used to explore the properties of current sheets at both 2D and 3D magnetic null points (Craig and Litvinenko 2005; Pontin and Craig 2005), in response to the kink instability (Craig and Sneyd 1990), and in various configurations conforming to the geometry of Parker’s braiding model—see the following section for details. It is worth noting that the method can fail if the computational grid becomes too distorted (Pontin et al. 2009), which has been somewhat mitigated in a new implementation by Candelaresi et al. (2014).

5.4 Current sheets at 3D magnetic nulls

Much like 2D magnetic nulls, 3D nulls are potential sites for current sheet formation. As in 2D (see Sect. 2.1), the Lorentz force at a 3D null tends to reinforce any perturbation to an equilibrium field, and, when the linear field about the null is considered in isolation, the current is shown to blow up in a finite time (Klapper et al. 1996; Bulanov and Sakai 1997). As in 2D, the energy that drives the collapse comes from the surrounding volume outside the modelled domain, so that the effect of the external conditions is being neglected in such models.

Antiochos (1996) was the first to sketch the form of the magnetic field when a discontinuity forms at a 3D null. The formation of such current sheets was first demonstrated explicitly by Pontin and Craig (2005) using ideal relaxation simulations (as in Sect. 5.3). They showed that any perturbation that disturbs the locations on the boundary of the footpoints of the spine or fan from their equilibrium positions leads to the formation of a current layer at the null in which the peak current shows a power-law divergence with the numerical resolution (Fig. 28c)—the expected signature of an underlying, unresolved current sheet. The divergence identically mirrors the behaviour for a 2D null (dashed curve in the figure) while the net flux of current in the sheet remains fixed with resolution in both cases.

Fig. 28
figure 28

(a, b) Spine (red) and fan (blue) structure in the final equilibrium obtained through an ideal relaxation simulation in which the spine footpoints are displaced, together with an isosurface (3D contour) of the current density (grey). The two different images show two different angles and illustrate the local collapse of the spine and fan. (c) Scaling of the peak current density \(J_{\max }\) and total current flux I with the numerical resolution for the 3D null (marked “\(k=0.5\)”) and for an equivalent simulation for a 2D null (marked “\(k=0\)”). Modified from Pontin and Craig (2005) with permission, copyright by AIP

The field line geometry takes the form of a local collapse of the spine towards the fan, directly analogous to the closing up of the separatrices when current sheets form at 2D nulls (Fig. 28a,b; see also Fuentes-Fernández and Parnell 2013). Fuentes-Fernández and Parnell (2012) considered ideal relaxation of non-equilibrium fields containing spiral nulls. In all but the exactly axisymmetric case they reported an infinite-time singularity of the current in a sheet that extends along the spine and weak-field fan direction (the spine and fan remain perpendicular to one another). More complex field geometries containing nulls have also been examined in ideal relaxation experiments, with signatures of singular current structures present at the nulls for a broad range of fields and perturbations (e.g., Pontin and Huang 2012; Craig and Pontin 2014; Candelaresi et al. 2015).

For current singularities at both 2D and 3D nulls, increasing the plasma pressure is found to weaken the divergence of the current with numerical resolution, but not to mitigate that divergence (Craig and Litvinenko 2005; Pontin and Craig 2005). This is consistent with the fact that locally around the null—where the field is linear—it can be shown that the \({\mathbf {j}}\times {\mathbf {B}}\) force never takes the form of a gradient, and therefore cannot be balanced locally by a pressure gradient (Parnell et al. 1997).

5.5 Line-tied magnetic fields without null points

5.5.1 Topological dissipation

As described in Sect. 5.15.3, when a magnetic field contains topologically distinct flux systems, partitioned by separatrices, an ideal evolution from one smooth equilibrium to another is not always possible and instead—in the absence of reconnection—current sheets form at null points and along separatrices. However, Parker (1972, 1979, 1989, 1994) went a step further by arguing that current sheets might also form during the evolution of magnetic fields without any separatrices, but with a complex winding of their magnetic field lines—and may therefore possibly contribute to coronal heating (Fig. 29).

Fig. 29
figure 29

The effect on a an initial field of b twisting and c braiding motions

He suggested that such a configuration cannot, in general, adjust to a new smooth force-free equilibrium in response to finite-amplitude footpoint motions, but should instead evolve towards a configuration containing tangential discontinuities of \({\mathbf {B}}\), or current sheets. The formation of these current sheets and subsequent rapid reconnection he called topological dissipation, since it is the field line topology (winding or “braiding”) that is responsible for the formation of the current sheets. Relentless motions of the photospheric footpoints of coronal field lines implies that the field is continually responding by reconnecting and converting magnetic energy into heat, which offers a way to heat the solar corona, especially active regions. Since it was proposed, this idea has stimulated substantial debate, with many different approaches used to attempt to prove or disprove the hypothesis. This became known as the “Parker problem”. Here we briefly summarise relevant results, and direct the reader to the review by Pontin and Hornig (2020) for more details.

5.5.2 Arrays of flux tubes

If a set of flux tubes is closely packed together and each is twisted in the same direction, then Parker (1979) realised that current sheets will form at the boundaries of the tubes. He considered a magnetic field (\(B_{x} = \partial A / \partial y, B_{y} = -\partial A /\partial x, B_{z} = {{\mathrm {constant}}}\)) in equilibrium such that

$$\begin{aligned} A=K \sin k_{x} x \, \sin k_{y} y. \end{aligned}$$

This rectangular array of twisted flux tubes has adjacent cells with opposite twist. If instead all the cells have the same sense of twist, they are not in equilibrium and form current sheets at their boundaries. A similar configuration was considered by Longcope and Strauss (1994), who studied the coalescence instability (Finn and Kaw 1977) between pairs of flux tubes with the same sense of twist. In the absence of line-tying of these flux tubes, the current sheet that forms between the tubes is singular. However, the line-tying provides an additional magnetic tension force that halts the collapse when the current layer thickness is still finite.

Another candidate mechanism that has been proposed to explain current sheet formation in the context of coronal heating and solar flares is the ideal kink instability (Kruskal et al. 1958; Hood and Priest 1979). Again, in the absence of line-tying, singular current sheets form, on resonant surfaces. In the line-tied case a current layer still forms at the radius that corresponds to the resonant surface. The thickness of this current layer scales inversely with the distance between the line-tied boundaries, and the growth-rate of the instability is reduced as described by Huang et al. (2010). However, the issue of current sheet formation in the nonlinear phase of the instability is not yet fully resolved.

5.5.3 Magnetic braids

In the original paper, Parker (1972) considered infinitesimal departures from a uniform field between parallel, perfectly conducting plates. He argued that, if the pattern of small-scale variations is not uniform along the large-scale field, then the field cannot be in magnetostatic equilibrium. In other words, equilibrium exists only if the field variations consist of a simple twist extending from one footpoint to another. However, van Ballegooijen (1985) pointed out an error in the calculation, and indeed argued that a smooth equilibrium should always be accessible following an infinitesimal perturbation to a uniform field (see also Sakurai and Levine 1981; Zweibel and Li 1987). This is consistent with the paper by Bineau (1972) who proved that smooth force-free fields (where \({\mathbf {j}}\times {\mathbf {B}}=0 \)) exist in the vicinity of the potential field (i.e., for small \(\alpha \), where \(\nabla \times {\mathbf {B}}=\alpha {\mathbf {B}}\)).

In spite of the above results, there have been numerous arguments put forward for the formation of tangential discontinuities of \({\mathbf {B}}\) in response to finite-amplitude perturbations. These include persuasive intuitive arguments based on the optical analogy (e.g., Parker 1994), as well as studies of “topologically untwisted” fields by Low (2006), Low and Flyer (2007), Janse and Low (2009). Nevertheless, none of these studies explicitly demonstrates formation of tangential discontinuities, and indeed counter-arguments have been made demonstrating that simple tangential discontinuities (in the form of single smooth surfaces) cannot form in response to smooth boundary motions (van Ballegooijen 1985; Longcope and Strauss 1994; Cowley et al. 1997).

Computational approaches to the Parker problem typically involve ideal relaxation simulations using the approach described in Sect. 5.3. In short, such approaches have not found any conclusive evidence for the formation of current sheets (e.g., Craig and Sneyd 2005; Wilmot-Smith et al. 2009b). What is, however, clear is that, as the boundary perturbations are progressively increased, the corresponding equilibrium in the domain contains current layers that are progressively thinner and more intense (van Ballegooijen 1988a, b; Mikić et al. 1989; Rappazzo and Parker 2013). To understand why this must generally be the case, consider the following intuitive argument (Pontin and Hornig 2015). In a force-free field (with \(\nabla \times {\mathbf {B}}=\alpha {\mathbf {B}}\)), \(\alpha \) is constant along magnetic field lines (since \({\mathbf {B}}\cdot \nabla \alpha =0\)). Thus, if the field line mapping between the two line-tied boundaries contains small perpendicular length scales then so must \(\alpha \) (assuming that it is not constant but rather varies between field lines). Now, \(\alpha ={\mathbf {j}}\cdot {\mathbf {B}}/{\mathbf {B}}\cdot {\mathbf {B}}=j_\Vert /|{\mathbf {B}}|\), or \(j_\Vert =\alpha |{\mathbf {B}}|\), so that \(j_\Vert \) must have the same perpendicular length scales as \(\alpha \) and the field line mapping. (The modifying factor of the field strength is approximately constant in the Parker problem geometry.) The same conclusion is reached for magnetic fields close to force-free equilibrium by considering the correlation length of \(\alpha \) along field lines—see Pontin et al. (2016). This was demonstrated explicitly by Pontin and Hornig (2015) who simulated the ideal relaxation of a set of braided fields (following the earlier approach of Wilmot-Smith et al. 2009b). In their model, the “twist parameter” k (see Fig. 30) determines the field complexity, with larger k corresponding to “more braided” fields. This can be quantified using the squashing factor Q (see Sect. 2.6), which exhibits progressively larger numbers of thinner layers with higher values of Q as the field becomes more braided (see also Wilmot-Smith et al. 2009a). As shown in Fig. 30, the equilibria for these magnetic braids contain both QSLs and current layers whose thickness scales exponentially—with the same exponent—with the twist parameter. The conclusion from these studies is that continual braiding of the coronal field lines may not lead to tangential discontinuities of \({\mathbf {B}}\) (singular current sheets), but nevertheless onset of reconnection is inevitable as the current layers become exponentially thinner and more intense as the field lines become more tangled.

Fig. 30
figure 30

The ideal relaxation of a braided field. Top: isosurface of the modulus \(|{\mathbf {j}}|\) of the current density at 60% of maximum. Middle: \(|{\mathbf {j}}|\) in the plane \(z=0\). Bottom: \(\log _{10}(Q)\) in the plane \(z=-24\). For twist parameter given by a \(k=0.5\), b \(k=0.6\), c \(k=0.7\). Image modified with permission from Pontin and Hornig (2015), copyright by AAS

5.6 Current sheet formation at hyperbolic flux tubes or quasi-separators

What kinds of motion encourage current sheets to form at a quasi-separator and the region that surrounds it, namely, a hyperbolic flux tube (HFT)? Démoulin et al. (1996a) conjectured that any footpoint motion would tend to do so, and this was followed by a series of numerical experiments that confirmed formation of concentrated currents for specific geometries (Inverarity and Titov 1997; Galsgaard et al. 2003b; Aulanier et al. 2005; De Moortel and Galsgaard 2006a). Later, Titov (2007) clarified the problem by showing how pinching motions are much more effective at concentrating currents than rotating motions.

Like separators, HFTs (or quasi-separators) are favourable magnetic structures for current sheet formation, because their field lines in a quadrupolar configuration connect regions of strong and weak photospheric magnetic field, which provides a favourable condition for pinching by a stagnation flow.

Consider a straightened-out HFT formed between four sunspots lying in two planes. Shearing displacements will either turn the flux tube or twist it and will produce at the midplane either a rotation or a stagnation flow. This will deform a Lagrangian grid in two different ways (Fig. 31) (Titov et al. 2003). For the case of “twist”, most of the grid distortion is in a narrow central region, and so the middle of the HFT will pinch to a strong current layer.

Fig. 31
figure 31

Image reproduced with permission from Titov (2007), copyright by CUP

Non-pinching (top) and pinching (bottom) deformations of an HFT in the midplane \(z=0\) (dashed) due to turning and twisting shearing motions, respectively, applied to the HFT footpoints.

The maximum current density (at the centre of the configuration) is

$$\begin{aligned} j_{z}^{*} =\frac{2}{\mu } \left( h+\frac{B_{\Vert }}{2 L}\right) \sinh \xi . \end{aligned}$$

where 2L is the distance between the planes, h and \(B_{\Vert }\) are the initial values of the transverse field gradient and longitudinal field at the centre, and \( \xi = V_{\mathrm{s}} t/ l_{\mathrm{sh}}\) is the dimensionless displacement of each sunspot moving with velocity \(V_{\mathrm{s}}\) and creating a shear region of a half-width \(l_{\mathrm{sh}}\). Thus, when \(\xi >1\), a stagnation-point flow at the centre of the HFT causes it to pinch and the current density to grow exponentially with spot displacement.

At large \(\xi \), the above kinematic estimate can be improved by relaxing the unbalanced stress in the current layer and allowing it to compress in the transverse direction to an approximately force-free state (Titov et al. 2003). The resulting central current density is

$$\begin{aligned} j_{z\, \mathrm{eq}}^{*} \simeq j_{z}^{*} \left[ 1 + e^{\xi } \left( 0.91\frac{h l_{\mathrm{sh}}}{B_{\Vert }} + 0.57 \frac{l_{\mathrm{sh}}}{L} \right) ^{2} \right] , \end{aligned}$$

which is larger than the kinematic value \( j_{z}^{*}\) by a factor that grows exponentially with the displacement \(\xi \) and with decreasing \(B_{\Vert }\). Effenberger et al. (2011) used an adaptive mesh code to study current accumulation in an HFT, observing extremely high current densities on the scale of the computational grid. In the limit \(B_{\Vert } \rightarrow 0\) we find \(j_{z\, \mathrm{eq}}^{*} \rightarrow \infty \), in agreement with the analysis of current accumulation at null points (Bulanov and Olshanetsky 1984; Priest and Titov 1996; Craig and Litvinenko 2005).

Interestingly, a strong current layer can form even in an initially uniform field, as can be seen by putting \(h=0\) in the above expressions (52) and (53). This is because the pair of twisting and shearing footpoint motions interlocks the coronal field lines and so forms an HFT even if none exists initially. Early numerical experiments on Parker braiding confirmed such an effect (see Mikić et al. 1989; Galsgaard and Nordlund 1996; Longbottom et al. 1998, and the preceding section).

6 Magnetic annihilation

6.1 A 1D current sheet with diffusion and advection

A closely related process to magnetic reconnection is magnetic annihilation, which refers to the inwards transport and cancellation of straight, oppositely-directed field lines in a current sheet (of infinite length). The variation of the magnetic field \(({\mathbf{B}})\) in time is described by the induction equation

$$\begin{aligned} \frac{\partial {\mathbf{B}} }{ \partial t} = {{\varvec{\nabla }}} \times ({{{\mathbf{v}} \times {\mathbf{B}}}}) + \eta \nabla ^{2} {\mathbf{B}}, \end{aligned}$$

due to the basic processes of advection of the magnetic field with the plasma and diffusion through the plasma. The ratio of advection to diffusion (i.e., of the two terms on the right-hand side of Eq. 54) for a length-scale \(l_{0}\) is the magnetic Reynolds number \(R_{m} = l_{0} V_{0} / \eta \).

For a 2D steady state with flow \({\mathbf {v}}= v_x\hat{\mathbf{x}}+v_y\hat{\mathbf{y}}\) and magnetic field \({\mathbf {B}}= B_x\hat{\mathbf{x}}+B_y\hat{\mathbf{y}}\), Eq. (54) integrates to Ohm’s Law

$$\begin{aligned} {{{\mathbf{E}} + {\mathbf{v}} \times {\mathbf{B}}}} = \eta {{\varvec{\nabla }} \times B}, \end{aligned}$$

where Faraday’s Law \(({{\varvec{\nabla }} \times \mathbf{E}}={\mathbf{0}})\) implies that the electric field \({\mathbf{E }} = E \, \hat{\mathbf{z}}\) is uniform. In most parts of the interior and atmosphere of the Sun, the advection dominates, so that the right-hand side is negligible. This holds, for instance, in the ideal region around a diffusion region and outside shock waves.

For steady-state reconnection, the other main MHD equation is the equation of motion under the influence of a pressure gradient and a \({\mathbf {j}}\times {\mathbf {B}}\)-force, namely,

$$\begin{aligned} \rho ({{{\mathbf{v}} \cdot {\varvec{\nabla }}}}) {\mathbf{v}} = -{\varvec{\nabla }} \left[ p+\frac{B^{2} }{2 \mu }\right] +\frac{({{\mathbf{B}} \cdot {\varvec{\nabla }}}) {\mathbf{B}}}{\mu }. \end{aligned}$$

When advection is negligible (\(R_m\ll 1\)) and the magnetic field \((B(x,t) \hat{\mathbf{y}})\) is one-dimensional, Eq. (54) reduces to a 1D diffusion equation

$$\begin{aligned} \frac{\partial B }{ \partial t} = \eta \, \frac{\partial ^{2} B }{ \partial x^{2}}, \end{aligned}$$

and so field variations on a scale \(l_{0}\) diffuse away in a time \(\tau _{d} = l_{0}^{2}/ \eta \) and with a speed \(v_{d} = \eta / l_{0}\).

For an initial magnetic field B(x, 0) the solution is

$$\begin{aligned} B(x,t) = \int G(x-x^{\prime },t)\ B(x^{\prime },0) \; dx^{\prime } \end{aligned}$$

where \(G(x-x^{\prime },t) =(4 \pi \eta t)^{-1/2} \exp [ - (x-x^{\prime })^{2} /(4 \eta t)] \) is the Green’s function.

For example, if initially there is an infinitesimally thin current sheet at the origin, so that the magnetic field is a step function \((B=B_{0}\) for \(x>0\) and \(B=-B_{0}\) for \(x<0\)), the solution becomes

$$\begin{aligned} B(x,t) = \frac{2B_{0} }{ \surd \pi } \int ^{x/\surd (4 \eta t)}_{0} e^{-u^{2}}du, \end{aligned}$$

and the steep magnetic gradient at \(x=0\) spreads out in time, as shown in Fig. 32. The width (\(4\sqrt{\eta t}\)) of the sheet increases in time, and the field is said to be annihilated, since the field strength at a fixed position decreases in time. During this process, the field lines diffuse inwards through the plasma and cancel at \(x=0\), while the magnetic energy is converted into heat by ohmic dissipation.

Fig. 32
figure 32

The magnetic profile at three times (\(t=0\), \(t_{1}\), \(t_{2}\)) during its diffusion from an initial 1D step function (see Eq. 58)

In the opposite limit (\(R_{m} \gg 1\)) when advection dominates over diffusion, the magnetic field lines are frozen into the plasma. For example, the effect of a stagnation-point flow \(v_{x} =-V_{0}x / a\), \(v_{y} = V_{0} y / a\) on a 1D field \((B(x,t) \hat{\mathbf{y}})\) is to carry the field lines inwards from the sides and accumulate them near \(x=0\). Here the induction equation (54) becomes

$$\begin{aligned} \frac{\partial B }{ \partial t} - \frac{V_{0}x }{ a} \frac{\partial B }{ \partial x} = \frac{V_{0} B }{ a}, \end{aligned}$$

which may be solved by the method of characteristics.

6.2 Stagnation-point flow model

As shown above, as the field lines in a current sheet diffuse inwards and cancel, the sheet naturally tends to diffusively broaden. However, a steady state can be maintained if this outwards diffusion is balanced by an inwards transport of the magnetic field \((B(x) \hat{\mathbf{y}})\) by a stagnation-point flow (\(v_{x} =-V_{0}x/a, v_{y} = V_{0} y/a\)), where \(V_{0}/a\) is constant (Fig. 33). When \(R_{m} \gg 1\), an extremely thin current sheet is created with a small width (l) and thus a large magnetic gradient (\({\varvec{\nabla }} B\)) and current (\(j\sim B/(\mu l)\)) (Parker 1973; Sonnerup and Priest 1975).

Fig. 33
figure 33

a Steady-state model for magnetic annihilation in which straight magnetic field lines (solid lines) are carried in from both sides by a stagnation-point flow (red dashed). b The magnetic field B as a function of distance x (see Eq. 61), with the approximations when \(x\ll l\) and \(x\gg l\) shown dashed

In this case Ohm’s Law (55) with \(E={\mathrm {const}}\) becomes

$$\begin{aligned} E- \frac{V_{0}x }{ a} B= \eta \, \frac{dB }{dx}, \end{aligned}$$

whose solution is

$$\begin{aligned} B= \frac{2E_{0}a }{ V_{0} l} \, \exp \left( -\frac{x^{2} }{ l^{2}} \right) \ \int _{0}^{x / l} \exp (X^{2})\ dX, \end{aligned}$$

where \(l^{2} = V_{0} /(2 \eta a)\), as shown in Fig. 33. When \(x\gg l\), the right side of Eq. (60) is negligible, the field is frozen to the plasma and \(B \approx (Ea)/(V_{0}x)\). On the other hand, when \(x\ll l\), the second term on the left is negligible, the field lines diffuse through the plasma and \(B \approx Ex/ \eta \). This represents one of the few exact nonlinear solutions of MHD, and it has been generalised to a 3D stagnation-point flow, with a field \(({\mathbf{B}}(x))\) that rotates as it is carried in (Sonnerup and Priest 1975).

6.3 Time-dependent current sheet

The purely one-dimensional behaviour of a current sheet involving a magnetic field \(B(x,t) \hat{\mathbf{y}}\), plasma flow \(v(x,t) \hat{\mathbf{x}}\), density (\(\rho (x,t)\)) and pressure (p(xt)) has been described by Priest and Raadu (1975) using self-similar solutions for highly subsonic and sub-Alfvénic flows. Numerical computations have been carried out by Forbes et al. (1982) for the evolution of a current sheet when the magnetic diffusivity is suddenly enhanced.

At small times (Fig. 34(left)), within the diffusion region one sees the outwards propagation of a shock wave and an inflow of plasma, as well as a second shock that overtakes the first shock, coalescing with it. At large distances from the diffusion region (Fig. 34(right)), a magnetoacoustic wave pulse propagates outwards, consisting of a rarefaction followed by a compression (see also Takeshige et al. 2015). It turns out that this 1D model describes well the time-dependent formation of a current sheet by 2D null collapse, see Sect. 10.3.

Fig. 34
figure 34

Image reproduced with permission from Forbes et al. (1982), copyright by CUP

Evolution of a current sheet in response to a sudden increase in diffusivity: (left) the density at small times when the shock is within the diffusion region and \(\beta _{\infty } = 0.1\) is the plasma beta at large distances; (right) density \((\rho )\), mass flux \((\rho v)\) and magnetic field (B) for \(\beta _{\infty } = 0.2\) at large times when the waves are outside the diffusion region.

6.4 Reconnective annihilation

The stagnation-point solution (Sect. 6.2), with inflow along the x-axis, namely,

$$\begin{aligned} v_{x}=-x, \qquad v_{y} = y, \qquad {\mathbf{B}} = B_{y} (x) \, \hat{\mathbf{y}}, \end{aligned}$$

has been generalised by Craig and Henton (1995) (Fig. 35) by superposing a 1D term \((G(x) \hat{\mathbf{y}})\) and a magnetic field \((\lambda {\mathbf{v}})\) that is parallel to a flow \(v_{x} = -x, v_{y} = y -F(x)\), to give

$$\begin{aligned} {\mathbf {B}}=\lambda {\mathbf {v}}+ G(x) \hat{\mathbf{y}}, \end{aligned}$$

where \(\lambda \) is a constant and the functions F(x) and G(x) are determined by the equations of induction and motion. The parameter \(\lambda \) measures the departure of the solution from simple magnetic annihilation, while the plasma velocity in this exact solution of the incompressible MHD equations now consists of a stagnation-point flow plus a shear flow.

Fig. 35
figure 35

Image reproduced with permission from Priest (2014), copyright by CUP

The Craig–Henton reconnective annihilation model, showing streamlines (dashed) and magnetic field lines (solid) for \(\lambda = 0.9\). The one-dimensional diffusion region (shaded) extends to infinity in the positive and negative y-directions.

The process is known as reconnective annihilation (or magnetic merging) since it is closer in spirit to annihilation than to reconnection, with the 1D current sheet extending to infinity along the y-axis and both advection and diffusion being one-dimensional in nature. The width of the current sheet (like magnetic annihilation) scales with magnetic diffusivity like \(\eta ^{1/2}\).

Craig et al. (1995) and Craig and Fabling (1996) later extended the 2D solution to 3D, with a background 3D magnetic null point and stagnation-point flow. The authors present two solutions that involve current layers that are localised to either the spine or the fan. These are constructed by combining straight field lines (extending to infinity) with a background potential magnetic null point field using a construction as in Eq. (62). In each case the straight field lines are in oppositely-directed bundles localised to either the spine or fan, and form a tube of current around the spine (“spine reconnective annihilation”) or a slab of current around the fan (“fan reconnective annihilation”). The solutions containing a planar current layer in the fan were demonstrated to be accessible through a dynamic evolution in an incompressible plasma by Craig and Fabling (1998). However, a pressure gradient is required within the current sheet, which is larger for thinner current sheets, i.e., for smaller \(\eta \). This pressure gradient is a consequence of the restrictive but necessary choice of low-dimensionality disturbance fields: it is required to balance the Lorentz force within the current layer. It turns out that in a compressible plasma, such a pressure gradient cannot be maintained, and instead the magnetic field collapses, with the current layer becoming fully localised at the null rather than extending to infinity (Pontin et al. 2007a). By contrast, spine reconnective annihilation solutions appear not to be dynamically accessible (Titov et al. 2004; Pontin et al. 2007a).

7 Steady 2D reconnection models

In many applications reconnection is quasi-steady in the sense that it changes its behaviour slowly over many Alfvén travel times. Furthermore, steady reconnection is easier to study than time-dependent reconnection, so the emphasis in the early history of the subject was on understanding the nature of steady 2D reconnection. In particular, slow Sweet–Parker reconnection (Sect. 7.1) was followed by the fast Petschek mechanism (Sect. 7.2) and by other types of fast reconnection (such as the Almost-Uniform family) that depend on the initial and boundary conditions (Sect. 7.3).

Fast reconnection is now the standard explanation for rapid energy release in the corona, but three possibilities arise, as mentioned in Sect. 1.1, namely, steady Petschek or Almost-Uniform reconnection, collisionless reconnection modified by the Hall effect, and impulsive bursty reconnection due to secondary tearing. However, in each of these three cases roughly the same maximum mean rate of reconnection is reached. See Sect. 9 for details of these three types of fast reconnection, both collisional and collisionless.

7.1 Sweet–Parker mechanism

The aim of early reconnection theory was to find the steady rate of reconnection, namely, the speed with which field lines may enter the reconnection site and have their connections to plasma elements changed. The first model by Sweet (1958a, b) and Parker (1957, 1963) modelled a diffusion layer of length (2L) stretching along the whole interface between opposing magnetic fields.

Equating the first and third terms in Ohm’s law (54) gives the magnetic diffusion time

$$\begin{aligned} \tau _{d} = \frac{L_0^{2}}{\eta } = 10^{-9}\, L_0^{2} \, T^{3/2}, \end{aligned}$$

with \(L_0\) in metres and T in degrees K. This is huge in practice: for instance, a typical coronal length-scale (\(L=10^{7}\) m) and temperature (\(T = 10^{6}\) K) yields a diffusion time of \(\tau _d = 10^{14}\) sec. Releasing magnetic energy in a solar flare or coronal heating event therefore needs the creation of intense current sheets with enormous magnetic gradients and a tiny sheet thickness.

7.1.1 The basic Sweet–Parker model (1958)

Sweet and Parker gave an order-of-magnitude treatment for a current sheet or diffusion region of length 2L and width 2l (Fig. 36), for which oppositely directed magnetic fields \(\pm B_{i}\) are carried in from both sides at a speed \(v_{i}\). In a steady state, this will be the same as the diffusion speed (\(\tau _d/l\)) with which the sheet tends to diffuse outward, namely,

$$\begin{aligned} v_{i} = \frac{\eta }{ l}. \end{aligned}$$
Fig. 36
figure 36

The notation for Sweet–Parker reconnection, with magnetic field lines (black) transported into a diffusion region (shaded grey) by a plasma flow (blue arrows)

Furthermore, conservation of mass implies that the rates at which plasma enters (at speed \(v_i\)) and leaves (at speed \(v_o\)) the sheet must be the same, so that, if the density is uniform,

$$\begin{aligned} L \, v_{i} = l \, v_{o}. \end{aligned}$$

However, if the plasma is accelerated along the current sheet by the \({\mathbf {j}}\times {\mathbf {B}}\)-force, the outflow speed is just the inflow Alfvén speed, namely,

$$\begin{aligned} v_{o} =v_{Ai} \equiv \frac{B_{i} }{ \sqrt{\mu \rho }}. \end{aligned}$$

After eliminating the width (l) between Eqns. (63) and (64), the reconnection rate becomes

$$\begin{aligned} v_{i} = \frac{v_{Ai} }{ R_{mi} \,^{1/2}} \quad \text {or, in dimensionless terms,} \quad M_{i}=\frac{1}{R_{mi}^{1/2}}, \end{aligned}$$


$$\begin{aligned} M_{i} \equiv \frac{v_{i}}{v_{Ai}} \qquad \text {and} \qquad R_{mi} \equiv \frac{L \, v_{Ai}}{\eta } \end{aligned}$$

are the inflow Alfvén Mach number and magnetic Reynolds number, respectively.

According to the above equations, the plasma is ejected from a sheet of width \(l = {L / R_{mi}\,^{1/2}}\) with a magnetic field strength \(B_{o} =B_{i} l/L= {B_{i} / R_{mi} \,^{1/2}}.\) Since \(R_{mi} \gg 1\), we therefore find that, as well as \(v_{i} \ll v_{Ai}\), the sheet width is much smaller than its length (\( l \ll L\)) and the outflow field is much smaller than the inflow field (\(B_{o} \ll B_{i}\)).

The Sweet–Parker mechanism has a sheet length (L) equal to the global external length-scale \((L_{e})\) and so \(R_{mi}\) becomes the global magnetic Reynolds number \( R_{me} = {L_{e} \, v_{Ae} / \eta }\). In practice \(R_{me} \gg 1\), and so the reconnection rate is very small. In the solar corona, for example, \(R_{me}\) is typically \(10^{8}\)\(10^{12}\), giving a reconnection rate \(10^{-4}\)\(10^{-6}\) of the Alfvén speed, which is much too slow to account for, say, a solar flare.

Priest (2014) describes three interesting aspects of Sweet–Parker reconnection. The first is that a consideration of energetics implies that half the inflowing magnetic energy is converted into thermal energy and half into kinetic energy, so creating two hot fast jets of plasma with equipartition between thermal and kinetic energy. The second aspect concerns the assumption that the plasma pressures at the neutral point (\(p_{N} = p_{i} + B_{i}^{2}/ (2 \mu )\)) and outflow \((p_{o})\) are the same, so that the plasma is accelerated from rest at the neutral point to \(v_o=v_{Ai}\) at the outflow, with pressure gradients along the sheet playing no role. However, when pressure gradients are included, the reconnection rate (66) is modified to

$$\begin{aligned} M_{i} = \frac{2^{1/4} \left( 1+ \frac{1 }{ 2} \beta _{i} (1-{p_{o}/ p_{i}}) \right) ^{1/4} }{ \surd R_{mi}}, \end{aligned}$$

where \(\beta _{i} = 2 \mu p_{i}/B_{i}^{2}\). Thus, when the outflow pressure exceeds the neutral point pressure \((p_{o}>p_{N}),\) the outflow slows \((v_{o} < v_{i})\) and the reconnection rate falls \((M_{i}< 1/ \surd R_{mi}).\) The third aspect concerns the effect of compressibility, which is to increase the reconnection rate (\(M_i\)) by a factor \((\rho _o/\rho _i)^{1/2}\) when \(\rho _o>\rho _i\).

7.2 Petschek mechanism

Petschek (1964) realised that reconnection could be much faster (in terms of the rate at which magnetic flux is brought into the diffusion region) if the Sweet–Parker diffusion region were much smaller and occupied only a small part (of length \(L\ll L_{e}\)) of the boundary (of length \(L_e\)) between opposing fields. He also analysed the external flow outside the diffusion region (Fig. 37a) and suggested that most of the energy conversion takes place at four slow-mode MHD shock waves; indeed, \(\frac{2}{5}\) of the inflowing magnetic energy is converted to heat and \(\frac{3}{5}\) to kinetic energy. Petschek’s maximum reconnection rate is typically a tenth or a hundredth of the Alfvén speed.

Fig. 37
figure 37

a For any fast reconnection regime, including Petschek’s mechanism: the magnetic field (\(B_{e}\)) at large distances \(L_{e}\) is brought in by a flow \(v_{e}\) towards a diffusion region (shaded) of dimensions 2l and 2L, where the inflow field and flow are \(B_{i}\) and \(v_{i}\), respectively. The plasma is heated and accelerated by four shock waves (red) and then expelled into two regions to left and right. b Notation for the analysis of the upper inflow region

At the inflow to the diffusion region, values such as \(v_{i}\) and \(B_{i}\) are denoted by a subscript i and their relationship to external values (such as \(v_{e}\) and \(B_{e}\)) at large distances \(L_e\) can be determined. Reconnection models then depend on the external reconnection rate \((M_{e} = v_{e}/v_{Ae})\) and the external magnetic Reynolds number \((R_{me} = L_{e} v_{Ae}/ \eta ).\) Fast reconnection here refers to reconnection whose rate \((M_{e})\) is much larger than the Sweet–Parker value (\(1/\sqrt{R_{me}}\)).

The external region around the diffusion region is then analysed in order to determine how \(M_{i}\) depends on \(M_{e}\). First of all, conservation of magnetic flux for a steady state \((v_{i} B_{i} = v_{e} B_{e})\) may be written in dimensionless terms as

$$\begin{aligned} \frac{M_{i} }{ M_{e}} = \frac{B_{e}^{2} }{ B_{i}^{2}}. \end{aligned}$$

Also, the Sweet–Parker relations (63) and (64) determine the dimensions of the diffusion region in dimensionless terms as

$$\begin{aligned} \frac{L }{ L_{e}} = \frac{1 }{ R_{me}} \frac{1 }{ M_{e}\,^{1/2}} \frac{1 }{ M_{i}\,^{3/2}}, \qquad \text {and} \qquad \frac{l }{ L_{e}} = \frac{1 }{ R_{me}} \frac{1 }{ M_{e}\,^{1/2}} \frac{1 }{ M_{i}\,^{1/2}}. \end{aligned}$$

Thus, after a model of the external region has determined \(B_{i}/B_{e}\), Eq. (68) gives \(M_{i}/M_{e}\) while the diffusion region dimensions follow from Eq. (69) in terms of \(M_{e}\) and \(R_{me}\) alone. We first apply this approach to Petschek’s model of the external region (Sect. 7.2.1), and then generalise it to a larger family of models known as Almost-Uniform reconnection (Sect. 7.3).

7.2.1 Petschek’s model: almost-uniform, potential reconnection

In this context, the terms “potential”, “nonpotential”, “uniform” and “nonuniform” denote the magnetic field behaviour in the inflow region upstream of the diffusion region. Petschek’s model is “almost-uniform” because the field in the inflow region is a weakly curved perturbation to a uniform field \((B_{e})\), and it is “potential” because there is no current in the inflow region. It possesses four slow-mode shock waves that stand in the flow and accelerate plasma to the Alfvén speed \((v_{A})\) parallel to the shock front (Fig. 37a). In the upper inflow region, the upstream plasma is moving downwards at the same speed \((v_{s})\) as the shock is trying to propagate upwards, namely, \(v_{s} = B_{N}/ \surd (\mu \rho )\), where \(B_{N}\) is the normal field component, and thus a steady state is maintained.

The magnetic field decreases from a uniform value \((B_{e})\) at large distances to \(B_{i}\) at the inflow to the diffusion region, while the flow speed increases from \(v_{e}\) to \(v_{i}\). The shocks provide a normal field component \((B_{N})\) which causes a small distortion in the inflow field from the uniform value \((B_{e})\) at large distances. The field is therefore the sum of a uniform horizontal field \((B_{e} \hat{\mathbf{x}})\) and the field obtained by solving Laplace’s equation in the upper half plane with a normal component (\(B_{N}\)) imposed along the shock waves and vanishing at the diffusion region. To lowest order, the shock inclination is neglected, apart from providing \(B_{y}=2B_{N}\) on the x-axis between L and \(L_{e}\) and \(-2B_{N}\) between \(-L_{e}\) and \(-L\) (Fig. 37b). The resulting solution of Laplace’s equation gives a field at the diffusion-region inflow of

$$\begin{aligned} B_{i} = B_{e} -\frac{4B_{N} }{ \pi } \log \frac{L_{e} }{ L}, \end{aligned}$$

or, after writing \(B_{N}/\sqrt{\mu \rho } = v_{e}\) from the shock relations,

$$\begin{aligned} B_{i} = B_{e} \left( 1-\frac{4M_{e} }{ \pi } \log \frac{L_{e} }{ L} \right) . \end{aligned}$$

Petschek found that the diffusion region size decreases and the shock angle increases as the reconnection rate \((M_{e})\) increases. He suggested that the mechanism chokes itself off when \(M_e\) is too large, and, by putting \(B_{i} = \frac{1 }{ 2} B_{e}\), he estimated a maximum reconnection rate \((M_{e}^{*})\) of

$$\begin{aligned} M_{e}^{*} \approx \frac{\pi }{ 8 \log R_{me}}, \end{aligned}$$

which in practice is much faster than Sweet–Parker and lies between 0.1 and 0.01.

7.2.2 Is fast Petschek reconnection possible with near-uniform resistivity?

When a spatially nonuniform resistivity is employed that is enhanced around the X-point, then numerical MHD simulations can produce Petschek and other fast regimes of reconnection. However, they do not produce a Petschek configuration when the resistivity is spatially uniform (Scholer 1989; Biskamp 1986; Yan et al. 1992). This produced some doubts as to the validity of Petschek reconnection (Biskamp 1993; Kulsrud 2001), but these have now been dispelled for three reasons:

  1. (i)

    In analytical treatments of Petschek’s mechanism, it is of course impossible to match the diffusion region and external region mathematically, since this is a highly complex fully two-dimensional nonlinear set of resistive MHD partial differential equations. But it is possible to match the resistive internal region and ideal external region using average properties of the diffusion region (Soward and Priest 1977; Priest and Forbes 1986).

  2. (ii)

    In many examples of reconnection in the solar atmosphere, the physically relevant case is that in which the diffusion region may well have an enhanced resistivity due to for example current-induced microinstabilities, although this does need to be established by a full analysis (Sect. 9.3).

  3. (iii)

    Recent carefully designed numerical experiments (Baty et al. 2006, 2009b, a) have shown that it is possible to set up and maintain a Petschek-like solution when a quasi-uniform resistivity is adopted, as summarised below. These suggest that a truly uniform resistivity is likely to be marginally stable and disrupted in practice by, for example, secondary tearing.

First of all, Baty et al. (2006) developed a helpful procedure for setting up a genuine Petschek solution with enhanced resistivity by overspecifying the boundary conditions, as follows. The initial state of their simulations is a one-dimensional current sheet with magnetic field

$$\begin{aligned} {\mathbf {B}}=B_{0}\tanh (x/a)\ {\hat{{\mathbf {y}}}}, \end{aligned}$$

together with density and pressure profiles in static isothermal equilibrium. The usual time-dependent resistive MHD equations are solved in the first quadrant \(0\leqslant x\leqslant L_{x}\), \(0\leqslant y\leqslant L_{y}\), with symmetry conditions along \(x=0\) and \(y=0\). Each simulation has a resistivity profile

$$\begin{aligned} \eta (x)=(\eta _{0}-\eta _{1})\exp [-(x/l_{x})^{2}-(y/l_{y})^{2}]+\eta _{1}. \end{aligned}$$

Petschek reconnection is a particular steady-state reconnection solution (Priest and Forbes 1986; Forbes and Priest 1987; Forbes 2001). It represents an undriven case in which the characteristics propagate information away from the X-point rather than towards it, and so it cannot easily be obtained by driving a flow with external forcing, since that is much more likely in practice to lead to one of the other reconnection regimes such as flux pileup (Priest and Forbes 1986). On the other hand, using a locally enhanced resistivity with free inflow and outflow conditions can produce a Petschek state (Scholer 1989).

What Baty et al. (2006) did was to use a much simpler procedure, namely, to overspecify the system by using fixed boundary conditions at the inflow boundary \(x=L_{x}\) by imposing \(\rho \), \(v_{x}\), \(v_{y}\), \(B_{y}\) and total energy. Free boundary conditions are adopted at the outflow boundary \(y=L_{y}\). Then the system chooses its own inflow velocity (\(v_{e}\)) (and therefore reconnection rate \(M_{e}\)) that is different from the value \(v_{x}=v^{*}_{e}\) being imposed. In a narrow boundary layer near \(x=L_{x}\) the velocity changes from \(v^{*}_{e}\) to \(v_{e}\). The resulting magnetic field (Fig. 38a) has all the features of a Petschek solution. Different solutions may be obtained by varying the value of \(l_{y}\) and they agree well with the analytical solutions of Priest and Forbes (1986) for \(M_{e}(M_{i})\).

Fig. 38
figure 38

Image reproduced with permission from Baty et al. (2009b), copyright by AIP

a The magnetic field and current density structures for a numerical simulation of Petschek reconnection. Image reproduced with permission from Baty et al. (2006), copyright by AIP. b Magnetic field and current density structures for the final steady state with an asymmetric resistivity having a uniform resistivity of \(\eta _{0}=10^{-3}\) in the lower half-plane.

Baty et al. (2006) considered the effect of varying the parameter \(\eta _1\) (for fixed \(\eta _0\)). They concluded that a quasi-uniform resistivity profile with an extremely small negative gradient that dominates the inevitable numerical contribution near the X-point can produce a stable Petschek solution and that a truly uniform profile is probably marginally stable. Petschek’s model and other fast reconnection regimes are therefore valid when the resistivity is enhanced or close to uniform. Indeed, one ripple of a resistivity fluctuation of small amplitude is sufficient to seed the Petschek mode.

Later, Baty et al. (2009b) carried out new numerical experiments using the same setup as before but over a domain centred at the origin without symmetry conditions on the axes. A uniform resistivity was adopted in the lower half-plane (\(y\leqslant 0\)) together with the resistivity profile from Eq. (73) in the upper half-plane (\(y\geqslant 0\)). The result is to produce a Petschek solution in the whole domain even though the resistivity is uniform in the lower half-plane and slightly nonuniform in the upper half-plane, as shown in Fig. 38(b) for \(\eta _{0}=10^{-3}\) and \(\eta _{1}=3 \times 10^{-5}\). Thus, Petschek reconnection with uniform resistivity in a half-plane is driven and maintained by Petschek reconnection with nonuniform resistivity in the other half-plane. Figure 38b presents the final steady state, in which the shocks in the lower half-plane are thicker due to the higher resistivity there. In this state, the inflow profiles lie between those associated with the large and small resistivities, as does the current density profile, which possesses a maximum amplitude that is shifted from the origin to a location in the upper half-plane. Furthermore, the X-point and stagnation point no longer coincide since the pressure gradient no longer vanishes at the X-point.

Subsequently, Baty et al. (2009a) were able to produce for the first time a fast Petschek solution with uniform resistivity in the whole domain. It was achieved by adopting a nonuniform viscosity profile and exhibited all the expected features of a classical Petschek solution, with two pairs of standing slow-mode shock waves attached to a central diffusion region and the inflowing plasma representing a weak fast-mode expansion. The diffusion region has a two-scale structure with an inner resistive region surrounded by a visco-resistive region.

7.2.3 Non-steady Petschek model

Semenov et al. (1983) and Heyn (1996) set up time-dependent solutions of Petschek type initiated by a localised resistivity increase in a pre-existing current sheet (Fig. 39a). Fast and slow magnetoacoustic waves are launched into the medium, and, in the incompressible case, the fast-mode waves propagate outwards instantaneously and set up an inflow towards the X-point. Unlike steady Petschek reconnection, the inflow is not uniform to lowest order but decreases with distance and vanishes at infinity. After reconnection stops, there is a switch-off phase with its effect propagating outwards, which is absent from steady-state solutions.

Fig. 39
figure 39

Image reproduced with permission from Priest and Forbes (2000), copyright by CUP

Semenov’s time-dependent model of Petschek-type reconnection with a initial oppositely directed magnetic field lines (solid) of a current sheet and b the evolution of the magnetic field and shocks (dashed).

Near the X-line the inflow is super-slow-magnetosonic, and so curved slow-mode shock pairs form and enclose the rear of a tear-drop-shaped outflow (Fig. 39b). The front of this region propagates along the x-axis at the ambient Alfvén speed \(v_{Ae}\), and so the external scale-length \((L_{e}= v_{Ae} t \)) increases linearly with time, while near the origin a steady-state Petschek solution is set up.

The MHD equations in the inflow region are linearised about the initial state using a small parameter \(\epsilon (t) = E^{*}(t) /(v_{Ae} B_{e})\ll 1, \) where \(E^{*}(t)\) is the electric field at the X-line and \(B_{e}\) is the ambient field. The inflow is current-free to first order in \(\epsilon (t)\), as in steady-state Petschek theory. The flux function A(xyt) is to first order a solution of Laplace’s equation \((\nabla ^{2}A = 0\)), and so the general solution in the inflow region is \(B_{x} = B_{e} + \partial A_{1} / \partial y , \quad B_{y} = - \partial A_{1} / \partial x, \) where

$$\begin{aligned} A_{1} = \frac{y }{ 2 \pi } \int ^{\infty }_{-\infty } \frac{A_{1} (x^{\prime }, 0, t) }{(x-x^{\prime })^{2}+y^{2}} dx^{\prime }. \end{aligned}$$

Here the function \(A_{1}(x^{\prime }, 0, t^{\prime })\) is determined by the slow-mode jump conditions to be

$$\begin{aligned} A_{1} (x^{\prime },0, t^{\prime }) = B_{0} |x^{\prime }|g(|x^{\prime }|-v_{Ae} t^{\prime })-B_{0} \int ^{|x^{\prime }|}_{0} g(\xi - v_{Ae} t^{\prime })d \xi , \end{aligned}$$

where \(g(x-v_{Ae}t) = \epsilon (t-x/v_{Ae})= -E^{*} (t-x/v_{Ae}) /(B_{0} v_{Ae}) \) is the normalised reconnection rate (\(\epsilon \)), which depends on the time-variation of the electric field at the X-line.

As in the steady Petschek analysis, the Sweet–Parker relations are invoked to find the maximum rate of reconnection, which, when \(\eta \) is held constant after an initial increase, becomes

$$\begin{aligned} \epsilon (t) = \frac{\pi }{ 4 \, {{\mathrm {ln}}} [ \epsilon ^{2}(t) R_{me}(t)]} \approx \frac{\pi }{ 4 \, {{\mathrm {ln}}} R_{me}(t)}, \end{aligned}$$

where \(R_{me}(t) = v_{Ae} L_{e}(t) / \eta _{ave} = (v^{2}_{Ae} / \eta _{ave}) t \) and \(\eta _{ave}\) is the average diffusivity in the diffusion region. Furthermore, the diffusion region dimensions

$$\begin{aligned} L&= \frac{L_{e} (t)}{R_{me} (t) \, \epsilon ^{2} (t)} = \frac{\eta _{ave}}{v_{Ae}} \left[ \frac{4\ {{\mathrm {ln}}} (v^{2}_{Ae} t/\eta )}{\pi }\right] ^{2} \\ l&=L_{e}(t) \epsilon (t) = \frac{\eta _{ave}}{v_{Ae}}\frac{4\ {{\mathrm {ln}}}(v^{2}_{Ae} t/\eta )}{\pi } \end{aligned}$$

grow logarithmically in time.

7.3 Other families of fast 2D reconnection

The form, value and number of the boundary conditions is of crucial importance when solving partial differential equations, since often much physics is incorporated in them. Petschek’s mechanism, which is almost-uniform and potential, has been generalised in two distinct ways by adopting different boundary conditions to give regimes of Almost-Uniform Reconnection, which is almost-uniform but nonpotential, and Nonuniform Reconnection, as reviewed in Priest and Forbes (2000).

Spontaneous reconnection that is initiated by some localised instability (such as the tearing mode) and is not influenced by distant magnetic fields requires free boundary conditions and tends to produce Potential Reconnection, which can be either Almost-uniform or Non-uniform, depending on the initial state. However, in contrast, Driven Reconnection depends on the nature of the driving and tends to give rise to Non-potential Reconnection with waves of current stemming from the driven boundaries.

Numerical experiments have confirmed that fast reconnection can indeed be produced, provided a locally enhanced magnetic diffusivity is present in the diffusion region, due to current-induced micro-instabilities (Priest and Forbes 2000; Priest 2014). Thus, fast reconnection is a prime candidate for rapid release of magnetic energy in the solar atmosphere. Collisionless and impulsive bursty reconnection is summarised in Sect. 9.

Fig. 40
figure 40

Image reproduced with permission from Priest and Forbes (2000), copyright by CUP

Notation for a the ideal inflow region of Almost-Uniform Reconnection, where subscripts e and i refer to (external) values at (\(0,L_{e}\)) and at the inflow to the diffusion region (shaded), respectively. b Reconnection rate (\(M_{e}=v_e/v_{Ae}\)) as a function of inflow Alfvén Mach number (\(M_{i}=v_i/v_{Ai})\) for various values of the parameter b.

The inflow region in Petschek’s mechanism has the character of a fast-mode expansion, in which the pressure and field strength decrease and the flow converges as the magnetic field approaches the diffusion region. Priest and Forbes (1986) decided to explore different types of inflow, since, unlike Petschek’s mechanism, some numerical reconnection experiments showed diverging flows and large pressure gradients. They solve the steady, two-dimensional, ideal, incompressible MHD equations to find the relation between the external and inflow Alfvén Mach numbers \((M_{e} = v_{e} / v_{Ae}\) and \(M_{i} =v_{i}/v_{Ai})\) at the top and bottom of the box in Fig. 40a, i.e., the global and local reconnection rates. Solutions are sought in powers of the global reconnection rate \((M_{e} \ll 1)\) that are a small perturbation to a uniform field \(({\mathbf{B}}_{e} = B_{e}\ \hat{\mathbf{x}})\), namely, \({\mathbf{B=B}}_{e}+M_{e}\ {\mathbf{B}}_{1}+ \ldots , \ \ {\mathbf{v}}=M_{e}\ {\mathbf{v}}_{1}+ \ldots ,\) where \((B_{1x},B_{1y})=(\partial A_{1}/\partial y,-\partial A_{1}/\partial x)\).

After calculating \(B_{i} / B_{e}\) and substituting into Eq. (68) in place of Petschek’s original expression (71), the graphs of \(M_{ e}\) against \(M_{ i}\) confirm that there is indeed a maximum reconnection rate \((M_{ e} \!^{*})\), as Petschek had suggested (Fig. 40b).

To lowest order in \(M_e\), the equation of motion becomes \( \nabla ^{2}A_{1} = - (\mu /B_{e})\) \(dp_{1}/dy, \) with a family of solutions having a rich diversity of properties

$$\begin{aligned} A_{1} = - \sum ^{\infty }_{0} \frac{a_{n} }{ (n+\frac{1 }{ 2}) \pi } \left\{ b - \cos \left[ \left( n+{\textstyle \frac{1 }{ 2}}\right) \pi \frac{x }{ L} \right] \right\} \cosh \left[ \left( n+{\textstyle \frac{1 }{ 2}}\right) \pi \left( 1-\frac{y }{ L}\right) \right] , \end{aligned}$$

which depend on a parameter b.

The reconnection rate \((M_{e})\) as a function of \(M_{i}\) and b for a given \(R_{me}\) is plotted in Fig. 40b. The type and rate of reconnection depend on the parameter b and therefore the inflow boundary conditions, since the horizontal flow at \((x,y)=(L_{e}, L_{e})\) is proportional to \((b-2/\pi )\). When \(b=0,\) Petschek’s solution is recovered (Fig. 41a). Although it is one particular member of a much wider class, it is special in the sense that it tends to occur for spontaneous rather than driven reconnection, since it is the only regime for which the fast mode characteristics are propagating away from the diffusion region. After calculating \(B_{i} / B_{e}\) and substituting into Eq. (68) in place of Petschek’s original expression (71), the graphs of \(M_{ e}\) against \(M_{ i}\) confirm that there is indeed a maximum reconnection rate \((M_{ e} \!^{*})\), as Petschek had suggested.

Fig. 41
figure 41

Image reproduced with permission from Priest and Forbes (2000), copyright by CUP

Almost-Uniform reconnection, showing two cases of the magnetic field lines (solid) and streamlines (dashed) in the upper half-plane.

Other values of b represent reconnection that is driven in various ways. When \(b<0\), near the y-axis the flow converges and so compresses the plasma by a slow-mode compression. When \(b>1\), the flow diverges and so expands the plasma by a slow-mode expansion; this type is known as the flux pile-up regime, since the field strength increases as the diffusion region is approached (Fig. 41b). When \(0<b<1\) a hybrid family of slow- and fast-mode expansions results. The central current sheet is much longer for the flux pile-up regime than the Petschek regime.

When the diffusion region becomes too long, it may become unstable to the secondary tearing or plasmoid instability (Sects. 8, 9.2) and a new regime of impulsive bursty reconnection results (Biskamp 1986; Priest 1986; Lee and Fu 1986b; Loureiro et al. 2007; Bhattacharjee et al. 2009). The almost-uniform theory has been compared with a variety of numerical experiments (Forbes and Priest 1987). It has also been generalised to give nonuniform regimes whose field lines possess a large curvature in the inflow region (Priest and Lee 1990).

8 Unsteady reconnection by resistive instability

Furth et al. (1963) realised that an equilibrium current sheet or sheared magnetic field can go unstable to resistive modes by reconnecting in a time-dependent way. The theory for one of them, the tearing mode, is given in Sect. 8.1, followed by extensions (Sect. 8.2), including cylindrical geometry, and nonlinear evolution (Sect. 8.2.1). In addition, when the diffusion region of a steadily reconnecting field becomes too long, it goes unstable to secondary tearing and an impulsive bursty regime of reconnection ensues (Priest 1986; Lee and Fu 1986a; Biskamp 1986; Forbes and Priest 1987); such secondary tearing has recently been studied extensively (Sect. 8.3).

Consider a current sheet or a sheared magnetic field, whose one-dimensional field is varying in the x-direction over a scale l with diffusion time \(\tau _{d} = l^{2} / \eta \) and Alfvén travel-time \(\tau _{A} = l/v_{A}\) across the field, such that \(\tau _{d} \gg \tau _{A}\). Furth et al. (1963) discovered that, when diffusion couples to magnetic forces, it can drive three resistive instabilities on time-scales \(\tau _{d}^{(1-{\lambda })}\tau _{A}^{\lambda }\), where \(0<\lambda < 1\), that are much faster than diffusion alone (on a time \(\tau _{d}\)). These instabilities create many small-scale magnetic islands in 2D or flux ropes in 3D, which later diffuse away and which may be important for coronal filamentation, diffusion and heating. The instabilities are called gravitational modes, rippling modes and tearing modes, with growth-rates \(\omega _{g}\), \(\omega _{r}\) and \(\omega _{tmi}\), respectively.

Gravitational modes are driven by gradients in density \((\rho _{0} (x))\), whereas rippling modes come from gradients in diffusivity \((\eta (x))\) that may be caused by a temperature gradient (Fig. 42a). Such modes have short wavelengths, of order the transverse scale \((kl \simeq 1)\), and so they create fine-scale filamentary structure in sheets and sheared fields with growth-rates

$$\begin{aligned} \omega _{g} = \left( \frac{(kl)^{2} \, \tau ^{2}_{A} }{ \tau _{d} \, \tau ^{4}_{G}} \right) ^{1/3}, \ \ \ \ \ \ \ \omega _{r} = \left[ \left( \frac{d \eta _{0} }{ dx} \frac{l }{ \eta _{0}} \right) ^{4} \frac{(kl)^{2} }{ \tau ^{3}_{d} \tau ^{2}_{A}} \right] ^{1/3}, \end{aligned}$$

where \(\tau _{G} = ( -g / \rho _{0} \, d \rho _{0}/ dx)^{-1/2}\) is the gravitational time-scale.

Fig. 42
figure 42

Image reproduced with permission from Priest and Forbes (2000), copyright by CUP

Magnetic field lines and plasma velocity (solid arrows) for resistive instabilities of a small-wavelength and b long-wavelength in a one-dimensional magnetic field (i.e., a current sheet or a sheared field), with x (or r in a cylindrical geometry) as the coordinate across the field.

The wavelength for tearing mode instability is very long, much greater than the current sheet width \((kl \ll 1)\), and so it can be more disruptive for a magnetic field than the other two modes (Fig. 42b). The growth-rate here is \(\omega = [\tau ^{3}_{d} \; \tau ^{2}_{A} \; (kl)^{2}]^{-1/5}\) for \((\tau _{A} / \tau _{d})^{1/4}<kl <1.\) The longest wavelength has the fastest growth-rate, namely,

$$\begin{aligned} \omega _{tmi} = \left( \frac{ 1 }{ \tau _{d} \; \tau _{A}}\right) ^{1/2}. \end{aligned}$$

Magnetic diffusion plays an important role only in a narrow region of width \(\epsilon l= (kl)^{-3/5} (\tau _{A} / \tau _{d})^{-2/5}l\). A perturbation such as in Fig. 42b, to a one-dimensional sheet with straight field lines, produces forces that make the perturbation grow. The magnetic tension tends to pull out the new loops of field away from the X-points along the sheet, while the magnetic pressure pushes plasma in from above and below towards them. The magnetic tension force due to large-scale curvature of the field lines produces a restoring force that is minimised for long wavelengths. The analysis of this instability is as follows.

8.1 Linear analysis of tearing-mode instability

Resistive instabilities can occur in any sheared magnetic field, not just a neutral current sheet, since the stability analysis is unaffected by the addition of a uniform field normal to the plane of Fig. 42. Sheared fields are, in general, resistively unstable at many thin sheaths throughout a structure. At any location the instabilities have a vector wavenumber \(({\mathbf{k}})\) perpendicular to the equilibrium field \(({\mathbf{B}}_{0})\), so that \( {{{\mathbf{k}} \cdot {\mathbf{B}}}}_{0}=0 \) and the crests of the perturbation lie in the plane of Fig. 42. Suppose the equilibrium plasma is at rest and the magnetic field has the form \({\mathbf{B}}_{0} = B_{0y} (x) \; \hat{\mathbf{y}} + B_{0z} (x) \; \hat{\mathbf{z}},\) with field lines that lie in yz-planes but rotate with x.

When the diffusivity (\(\eta \)) is uniform and the plasma incompressible, the MHD induction equation and the curl of the equation of motion become

$$\begin{aligned} \frac{\partial {\mathbf{B}} }{ \partial t}= & {} {\varvec{\nabla }} \times ({{{\mathbf{v}} \times {\mathbf{B}}}})+ \eta \nabla ^{2} {\mathbf{B}}, \end{aligned}$$
$$\begin{aligned} \rho \frac{d }{ dt} ({{\varvec{\nabla }} \times \mathbf{v}})= & {} {\varvec{\nabla }} \times \left[ \frac{({\varvec{\nabla }} \times {\mathbf{B}}) \times {\mathbf{B}}}{\mu }\right] . \end{aligned}$$

Small perturbations are made in the form

$$\begin{aligned} {\mathbf{v}}_{1} (x) \exp [i(k_{y} y+k_{z}z)+\omega t], \ \ \qquad {\mathbf{B}}_{1} (x) \exp [i(k_{y} y+k_{z} z) + \omega t], \end{aligned}$$

while \({{\varvec{\nabla }} \cdot {\mathbf{B}}} =0\) and \({{\varvec{\nabla }} \cdot \mathbf{v}} = 0\) are used to eliminate \(v_{1y}\) and \(B_{1y}\). Then, after nondimensionalising the variables (\(\bar{\mathbf{B}} = {\mathbf{B}}/B_{0}, \bar{\mathbf{v}}_{1}=-{\mathbf{v}}_{1} ikl^{2}/ \eta , {\bar{k}} = kl, \bar{\omega } = \omega l^{2}/ \eta , {\bar{x}} = x/l\)), the x-component of Eq. (74) and z-component of Eq. (75) become

$$\begin{aligned} \bar{\omega } {\bar{B}}_{1x}= & {} - {\bar{v}}_{1x} f+ ({\bar{B}}_{1x}^{\prime \prime } - \bar{k^{2}} {\bar{B}}_{1x}), \end{aligned}$$
$$\begin{aligned} \ \ \omega ({\bar{v}}^{\prime \prime }_{1x} - \bar{k^{2}} {\bar{v}}_{1x} )= & {} R_{m}^{2} \, \bar{k^{2}} f\left[ -{\bar{B}}_{1x} f^{\prime \prime } /f+( {\bar{B}}^{\prime \prime }_{1x} - \bar{k^{2}} {\bar{B}}_{1x})\right] , \end{aligned}$$

where a dash represents a derivative with respect to \({\bar{x}}\) and \(f={\mathbf{k}} \cdot \bar{\mathbf{B}}_{0}/k \).

When the magnetic Reynolds number is large (\(R_{m}= {lv_{A} / \eta } = {\tau _{d} /\tau _{A}} \gg 1\)), magnetic diffusion is negligible except in thin sheets (of width \(2 \epsilon l\), say) where \({{{\mathbf{k}} \cdot {\mathbf{B}}}}_{0} = kf=0\) and reconnection occurs. If the centre of the sheet is located at \(x=0\) and \(k_{z} =0\), then \({{{\mathbf{k}} \cdot {\mathbf{B}}}}_{0} =0\) reduces to \(B_{0y}(x) = 0\) or \(x=0\). Solutions to Eqns.(76) and (77) are then found in an outer region \((|{{{\bar{x}}}}|> \epsilon )\) and an inner region \((|{{{\bar{x}}}}|< \epsilon )\) and matched at the boundary \(({{{\bar{x}}}}= \epsilon )\) between them.

In the outer region diffusion is negligible and Eqns.(76) and (77) reduce to

$$\begin{aligned} \bar{\omega } {\bar{B}}_{1x}= & {} - {\bar{v}}_{1x} {\bar{B}}_{0y} + ( {\bar{B}}^{\prime \prime }_{1x}-\bar{k^{2}} {\bar{B}}_{1x}) ,\nonumber \\ 0= & {} - {\bar{B}}_{1x} {\bar{B}}^{\prime \prime }_{0y}/ {\bar{B}}_{0y} +({\bar{B}}^{\prime \prime }_{1x} - \bar{k^{2}} {\bar{B}}_{1x}), \end{aligned}$$

whose solution for \({\bar{B}}_{1x}\) depends on the equilibrium field \({\bar{B}}_{0y} ({\bar{x}})\). For example, a step-profile (\({\bar{B}}_{0y} = 1 \ {{\mathrm {for}}}\ {\bar{x}} >1,\ \ {\bar{x}}\ {{\mathrm {for}}} \ |{\bar{x}} |<1,\ \ {{\mathrm {and}}}\ -1\ {{\mathrm {for}}}\ {\bar{x}} <-1\)) leads to

$$\begin{aligned} {\bar{B}}_{1x} = \left\{ \begin{array}{ll} a_{1} \sinh {\bar{k}} {\bar{x}} + b_{1} \cosh {\bar{k}} {\bar{x}} \qquad &{} {\bar{x}} <1, \\ a_{0} \exp (- {\bar{k}} {\bar{x}}) &{} {\bar{x}} > 1, \end{array} \right. \end{aligned}$$

for \({{\bar{x}}} >0\), where conditions at \({\bar{x}} =1\) determine the constants \(a_{1}\) and \(b_{1}\) for \({\bar{x}} >0\). The corresponding solution for \({{\bar{x}}} <0\) has the same value of \(b_{1}\) but the opposite sign (same magnitude) of \(a_{1}\), which leads to a jump in the value of \({\bar{B}}^{\prime }_{1x}/ {\bar{B}}_{1x}\), known as delta prime, given by \({\varDelta }^{\prime } = [ {\bar{B}}^{\prime }_{1x} / {\bar{B}}_{1x} ]^{0+}_{0-} = 2 a_{1} {\bar{k}}/ b_{1}.\)

The inner region, in which diffusion is important, has a width of order \(\epsilon l\), where \(\epsilon ^{4} = {\bar{\omega } / (4 {\bar{k}}^{2} R_{m}^{2})}\). After defining inner variables \(X= {\bar{x}} / \epsilon , \, V_{1x} = {\bar{v}}_{1x} (4 \epsilon / \bar{\omega }),\) Eqns.(76) and (77) reduce to

$$\begin{aligned} \ddot{{\bar{B}}}_{1x}= & {} \epsilon ^{2} \bar{k^{2}} {\bar{B}}_{1x} + \epsilon ^{2} \bar{\omega } ( {\bar{B}}_{1x} + {\textstyle \frac{1 }{ 4}} V_{1x} X), \end{aligned}$$
$$\begin{aligned} {\ddot{V}}_{1x}= & {} V_{1x} (\bar{k^{2}} \epsilon ^{2} + {\textstyle \frac{1 }{ 4}} X^{2}) + {\bar{B}}_{1x} X, \end{aligned}$$

where dots represent X-derivatives. For long wavelengths \({\bar{k}} \ll 1\), Eq. (80) implies \(\ddot{{\bar{B}}}_{1x} \sim \epsilon ^{2} \, \bar{\omega } \, {\bar{B}}_{1x}\), so that \({\varDelta }^{\prime } = 2 ({{\bar{B}}^{\prime }_{1x} /{\bar{B}}_{1x}})_{x = \epsilon } = 2[ {\dot{{\bar{B}}}}_{1x} / (\epsilon {\bar{B}}_{1x} )]_{X=1} \sim \epsilon \, \bar{\omega }\).

If \({\bar{B}}_{1x}\) is assumed to be constant in the inner region (the so-called constant-psi approximation), Eq. (80) can be solved analytically and implies \({\varDelta }^{\prime } = 3 \, \epsilon \, \bar{\omega }\) when \({\bar{k}} \epsilon \ll 1\). Equating the two expressions for \({\varDelta }^{\prime }\) enables the outer and inner regions to be matched (Fig. 43) and gives a growth-rate of \(\bar{\omega } =[( 8R_{m})/ (9 {\bar{k}})]^{2/5}\), which lies between the diffusion \((\omega _{d}= \eta /l^{2})\) and Alfvén (\(\omega _{A} = v_{A}/l\)) rates. Eqns. (80) and (81) may also be solved when \({\bar{k}} \ll 1\) without assuming \({{{\bar{B}}}}_{1x}\) to be uniform in the internal region. The resulting dispersion relation has a growth-rate for the fastest-growing mode of \(\bar{\omega }_{\max } \simeq 0.6 \, R_{m}^{1/2} \) and a corresponding wave-number of \({\bar{k}}_{\max } \simeq 1.4 \,R_{m}^{-1/4}. \) In other words, the growth time (\(\tau _{m} = (\tau _{d} \, \tau _{A})^{1/2} \)) is the geometric mean of the diffusion and Alfvén times, so that long narrow islands are formed.

Fig. 43
figure 43

Image reproduced with permission from Priest and Forbes (2000), copyright by CUP

The tearing mode instability produces the magnetic field \(({\bar{B}}_{1x})\) and velocity \(({\bar{v}}_{1x})\) profiles shown, as functions of distance \(({\bar{x}}=x/l)\) normal to a current sheet, where l is the half-width of the sheet and \(\epsilon l\) is the half-width of the inner diffusive layer.

8.2 Extensions to the simple tearing-mode analysis

The above theory for tearing-mode instability has been extended in many different ways, which are reviewed in detail in Priest and Forbes (2000). These include the effects of different initial equilibrium profiles, the addition of a flow or an extra magnetic field component to the initial state, and a viscous force. Also, tearing may be driven faster by fast magnetoacoustic waves or small-scale MHD turbulence.

Ideal and resistive instabilities of a curved magnetic flux tube (major radius R and minor radius \(a\ll R\)) with both poloidal \(B_{p}(r)\) and toroidal \(B_{\phi }(r)\) components that vary with distance (r) from the axis, have been studied at length (e.g., Furth et al. 1973; Bateman 1978; Wesson 1997). Resistive modes can allow the magnetic field to slip through the plasma in a narrow layer around a resonant surface where \({{{\mathbf{k}} \cdot {\mathbf{B}}}} = 0\). Radial perturbations of the form \({{{{\varvec{\xi }} = {\varvec{\xi }}}}} (r) \exp [i(m \theta - n \phi )]\) are studied and a key parameter is the amount [\({\varPhi }_{T} (r) = {2 \pi \, R \, B_{p} /( r \, B_{\phi })}\)] by which a field line is twisted around the axis as it goes from one end of a flux tube to the other. A related quantity is the safety factor \(q(r) = {2 \pi /{\varPhi }_{T}} \), namely, the number of turns that a field line makes around the major axis during one turn around the minor axis of a torus.

A flux tube has three ideal modes. m=1 kink modes arise when the twist is too large and are driven by the current gradient. Internal interchange modes are driven by a pressure gradient. A ballooning mode is driven by pressure gradients when the large-scale curvature of a torus is included.

When resistivity is included, the kink mode becomes a tearing mode when \(q_{a} >m\), with the resonant surface inside the tube and a growth-rate \(\tau _{d}^{-3/5} \tau _{A}^{-2/5}\). Also, an internal mode with \(m>1\) becomes a resistive interchange mode, with a growth-rate of \(\tau _{d}^{-1/3} \tau _{A}^{-2/3}\), while increased pressure gradients produce resistive ballooning modes (e.g., Strauss 1981). The \(m=1\) internal resistive kink mode becomes unstable when \(q_{0}<1\). The effect of increasing the twist of a flux tube is therefore to introduce the tearing-mode first and then later the ideal kink mode.

For a coronal loop or solar flare with a twist \(({\varPhi }_{T} =LB_{p} /(rB_{\phi }))\), an important feature is the presence of line-tying in the dense photosphere at the feet of the coronal magnetic fields. For the kink instability in a flux tube of uniform twist, the threshold for kink instability is \({\varPhi }_{T} =2.5 \pi \) (Hood and Priest 1981). Its nonlinear development can make the flux tube highly kinked, with many reconnecting current sheets (see Sect. 14.2). However, the effect of a resistive kink instability with its lower threshold for a coronal loop with line-tying has not yet been explored, as far as we are aware.

8.2.1 Nonlinear evolution of tearing

The nonlinear development of tearing can proceed in different ways, depending on the magnetic Reynolds number \(R_{m}\), and the form of the equilibrium and boundary conditions. It may just saturate nonlinearly or, especially in the solar atmosphere, it may develop rapidly, as detailed in Priest and Forbes (2000). We just give a brief summary here.

One possible pathway is for the instability at constant-psi to saturate nonlinearly (Rutherford 1973) due to inwards diffusion from the external region to produce wider magnetic islands and to slow the rate of energy release, eventually saturating at an island width of order \({\varDelta }^{\prime }_{1} (0) l^{2}\) (White et al. 1977). Numerical confirmation for \(m=2\) has been presented (Park et al. 1984). However, \(m=1\) tearing is very different, since \(\psi \) is no longer constant in the island. A heuristic argument by Kadomtsev (1975) suggested that reconnection continues to grow until the current density flattens off inside the \(q=1\) surface, which was confirmed in a numerical study by Schnack and Killeen (1979) (Fig. 44). A large magnetic island grows rapidly in the nonlinear phase and then slow reconnection eats it away and leads to a new set of nested surfaces with one O-point.

Fig. 44
figure 44

Image reproduced with permission from Priest and Forbes (2000), copyright by CUP, after Schnack and Killeen (1979)

Magnetic flux surfaces during the linear (\(t=10 \tau _{A}\)) and nonlinear \((t= 60 \tau _{A})\) phases of \(m=1\) resistive kink instability.

Another pathway is nonlinear mode coupling of different tearing modes (Waddell et al. 1976; Diamond et al. 1984), which can produce high-m turbulence that may extend across a significant fraction of the minor radius.

A third pathway is for a chain of magnetic islands to undergo an ideal coalescence instability (Finn and Kaw 1977; Longcope and Strauss 1993). The way in which nonlinear coalescence depends on island amplitude (\(\epsilon \)) and \(R_{m}\) has been studied by Pritchett and Wu (1979) and Biskamp (1982). In a numerical investigation of coalescence, Bhattacharjee et al. (1983) found that the islands first approach and form a current sheet, at which reconnection takes place, with the instability saturating after typically 30 \(\tau _{A}\) and leaving a single island oscillating in response to its dynamic evolution.

A fourth pathway, especially at the extremely large values of \(R_{m}\) expected in the solar atmosphere, is for a current sheet to grow so long that it goes unstable to secondary tearing and enters a regime of impulsive bursty reconnection, as described below in Sect. 8.3.

In a solar current sheet with a large \(R_{m}\) that is line-tied at one end to the photosphere, tearing may develop at the neutral point closest to the surface, which may well develop fast nonlinear Petschek reconnection and subsequently possibly undergo secondary tearing and coalescence.

8.3 Fast impulsive bursty reconnection via the plasmoid instability

Reconnection in practice is not steady but is “impulsive and bursty”, with reconnection rapidly switching on and off or changing between fast and slow in a quasi-periodic or random manner, as suggested by Priest (1986) and analysed by Biskamp (1986); Lee and Fu (1986b); Loureiro et al. (2007); Bhattacharjee et al. (2009). This may happen due to several scenarios, the first of which is the switching on or off of a turbulent magnetic diffusivity (\(\eta _{\mathrm{turb}}\)) as the current density exceeds or falls below the critical current (\(j_{\mathrm{crit}}\)) for the onset of microinstabilities. Other possibilities are that reconnection may switch on and off near a metastable state, as seen in some numerical experiments (Baty et al. 2009a), or that the nonlinear development of a resistive instability may lead to the formation, fragmentation and coalescence of magnetic islands. Indeed, in recent years there has been a resurgence of interest in this latter possibility, which we review in the remainder of this section.

In astrophysical plasmas, the extreme values of \(R_m\) mean that monolithic current layers provide very slow reconnection rates, at odds with the rapid energy release observed during, for example, solar flares. As described in Sect. 8.1, the tendency for such layers to fragment into a ‘chain’ of magnetic islands or plasmoids was established by Furth et al. (1963), although the growth-rate of the instability was initially thought to be too slow to explain fast reconnection onset.

8.3.1 Plasmoid instability in 2D planar current sheets

Building on earlier work by Bulanov et al. (1979), Biskamp (1986) suggested that a Sweet–Parker current sheet may be unstable for aspect ratios \(L/l>100\) (see also the discussion of Forbes and Priest 1987, and references therein). Our understanding of the importance of tearing for the onset of fast reconnection was then revolutionised by the discovery of what is now known as the “plasmoid instability”, starting with the work of Loureiro et al. (2007) (see also Tajima and Shibata 1997) who made a stability analysis of a finite-length, Sweet–Parker current sheet (i.e., a current sheet with aspect ratio \(l/L\sim S^{-1/2}\), where \(S=Lv_A/\eta \) is the Lundquist number; see Sect. 7.1). They solved the reduced MHD equations, matching the solutions in an “outer” ideal region and an “inner” region comprising the current layer. They discovered that the instability growth-rate (\(\gamma \)) for the fastest growing mode (namely, \(\gamma _{\max }\tau _A\sim S^{1/4}\)) increases with increasing S, where \(\tau _A=L/v_A\) is the Alfvén time-scale. The corresponding wavenumber k of the fastest growing mode scales as \(k_{\max }L\sim S^{3/8}\).

Following these initial analyses, the theory has been extended to include the effects of plasma viscosity (Loureiro et al. 2013; Comisso et al. 2017), as well as three-dimensional effects (Baalrud et al. 2012). The scalings have been subsequently confirmed in various computational simulations, primarily beginning from 1D initial states such as Harris sheet configurations (e.g., Bhattacharjee et al. 2009; Samtaney et al. 2009; Loureiro et al. 2013). The critical current sheet aspect ratio for onset of tearing has been confirmed in simulations to be of order 100 (e.g., Loureiro et al. 2005; Samtaney et al. 2009). The fact that the growth-rate scales positively with the Lundquist number presents an issue in the limit \(S\rightarrow \infty \), which has been considered by Pucci and Velli (2013) and Uzdensky and Loureiro (2016), who made slightly different arguments regarding saturation of the growth-rate for large S. An analogous instability with similar onset threshold values, for both S and the current sheet aspect ratio, was discovered for a current sheet formed by the collapse of a 3D null point by Wyper and Pontin (2014b).

The linear phase of the instability is characterised by the formation of plasmoids in the current layer. Due to the large growth-rate at large S, simulations rapidly access the nonlinear phase. It is in this nonlinear phase, when the plasmoids grow to sizes larger than the current sheet width, that the reconnection process becomes substantially affected by the plasmoid instability dynamics. Here the evolution becomes highly dynamic and “bursty”, involving, for example, the nonlinear growth of plasmoids, their ejection from the current layer, coalescence of plasmoids, and associated “secondary tearing” of the current sheets that mediate this coalescence (see Fig. 45). This leads to a “fractal-like” structure for current sheets on a hierarchy of length scales (Shibata and Tanuma 2001).

Fig. 45
figure 45

Image reproduced with permission from Bhattacharjee et al. (2009), copyright by AIP

Time evolution of 2D plasmoid instability simulations, showing the component of the current density out of the 2D plane (colour scale) and magnetic field lines (black), for a Lundquist number \(6.28\times 10^5\).

While the reconnection rate in the nonlinear phase is found to fluctuate greatly in time, when these fluctuations are averaged, the overall reconnection rate (determined in 2D by the electric field at the “dominant” X-point) becomes nearly independent of the resistivity (e.g., Bhattacharjee et al. 2009; Loureiro et al. 2012). It is argued that this is because the reconnection rate can be approximated by applying the Sweet–Parker model to the smallest current sheets in the hierarchy, for which the reconnection rate scales as \(\sqrt{S_c}\), where \(S_c\) is the critical Lundquist number, and is thus independent of the global Lundquist number (Uzdensky et al. 2010).

The statistical properties of the hierarchy of plasmoids that forms after the nonlinear stage has fully developed is important for describing the long-time dynamics. As such, a number of studies have addressed this plasmoid “spectrum” in some detail. Uzdensky et al. (2010) considered a typical plasmoid in some arbitrary level of the current sheet hierarchy, formed away from the current sheet centre, and ejected in the outflow while growing in size and flux due to the reconnection process. They showed that the distribution function for plasmoids of flux \({\varPsi }\) should be \(f({\varPsi })\sim {\varPsi }^{-2}\), while their size (w) distribution should be \(f(w)\sim w^{-2}\). In numerical simulations, the picture is found to be a little more complicated, with at least parts of the distribution of plasmoid fluxes and sizes having an exponent closer to \(-1\) than \(-2\), and physical explanations have been proposed to account for these discrepancies (Huang and Bhattacharjee 2012; Loureiro et al. 2012).

Also predicted by Uzdensky et al. (2010) is the formation of rare “monster plasmoids”, which are created when an island begins to grow by chance close to the centre of a current layer; such plasmoids are not entrained in a strong current layer outflow, and so are able to grow to “monster” sizes before being ejected from the current layer to coalesce with plasmoids further up the hierarchy. The formation of such monster plasmoids is confirmed in simulations (Loureiro et al. 2012; Nemati et al. 2017), and it is proposed that they may be associated with large, violent bursts during reconnection events.

8.3.2 Plasmoid instability in more complex geometries and in observations

The above analyses and simulations have focussed solely on the 2D case, while the nonlinear evolution in 3D has been much less explored. In 3D, the range of current sheet configurations that may be susceptible to a plasmoid-type instability is much richer. The simplest way to extend into 3D is to include a field component in the third dimension and to model a fully 3D domain. This was done by Huang and Bhattacharjee (2016) who simulated the coalescence instability in 3D. They observed that, following the initial tearing of the current sheet and formation of plasmoid-like structures, the reconnection layer quickly transitioned to a turbulent behaviour with fewer well-defined plasmoids (in cross-sections) than comparable 2D simulations—see Fig. 46.

Fig. 46
figure 46

Image reproduced with permission from Huang and Bhattacharjee (2016), copyright by AAS

Current density and magnetic field lines in a 3D MHD simulation of the plasmoid instability during coalescence instability with an initially uniform guide field.

This transition to turbulence is also present in the picture developed by Dahlburg and Einaudi (2002); Dahlburg et al. (2005) who considered tearing in a sheared magnetic field (though there are differences in the detailed interpretation). In their simulations, the twisted flux tubes formed by the initial tearing undergo a “secondary instability”—essentially an ideal kinking of those twisted flux tubes—for sufficiently large shear across the current layer, leading subsequently to a turbulent evolution. This secondary instability was proposed as an explanation for the “switch-on” nature of coronal heating events in Parker’s braiding picture of coronal heating.

More recently, this has been revisited by Leake et al. (2020) who undertook a set of simulations for different current sheet lengths and magnetic shear angles. They discovered two different regimes of behaviour, distinguished by the relative values of the current sheet length \(L_s\) and the wavelength \(\lambda _f\) of the fastest growing parallel mode (in the corresponding infinite system). For \(L_s>\lambda _f\), sub-harmonics of the fastest-growing parallel modes are present, and the nonlinear interaction of these sub-harmonics and the subsequent coalescence of 3D plasmoids dominates in the nonlinear phase. By contrast, for \(L_s<\lambda _f\), the fastest growing parallel mode has no sub-harmonics, and rapid energy conversion only takes place when the magnetic shear is large, in which case oblique modes grow large enough to interact nonlinearly with the dominant parallel mode. There are discrepancies in the interpretation of the nonlinear behaviour with those of Dahlburg and Einaudi (2002); Dahlburg et al. (2005), and it is clear that further exploration is required. Moreover, in order to determine the relevance for energy release in coronal current sheets, the stabilising effect of photospheric line-tying must in future be included.

A nonlinear tearing or plasmoid-type instability has also been discovered in a current sheet formed by the collapse of a 3D magnetic null by Wyper and Pontin (2014b, 2014a). The current sheet was formed by shear driving of the spine field lines as shown in the upper panel of Fig. 47. Tearing onset was observed to occur when the current sheet aspect ratio exceeds around 50, and the nonlinear dynamics involves the formation of a plethora of 3D nulls due to multiple bifurcations (see Sect. 2.4) and accompanying twisted flux ropes. By contrast with plasmoids in 2D these flux ropes are not enclosed by magnetic flux surfaces, and field lines intertwine between multiple flux ropes as the nonlinear phase develops (Fig. 47). Since the magnetic shear angle goes from \(180^\circ \) at the null to small values further from the null, a detailed understanding of the dynamics in terms of perturbation modes is a formidable challenge.

Fig. 47
figure 47

Image reproduced with permission from Wyper and Pontin (2014a), copyright by AIP

Current density and magnetic field lines in a 3D MHD simulation of the plasmoid instability when a 3D magnetic null point is subjected to continuous shearing motions on the spine boundaries, which drives spine-fan reconnection (see Sect. 10.2.1).

Observations and simulations of dynamic events in various layers of the solar atmosphere are providing evidence for the importance of the plasmoid instability for energy release on the Sun. Rouppe van der Voort et al. (2017) present observations of bursty energy release in UV bursts in the chromosphere. These are compared with a reconnection event that was identified in a 2.5D radiative MHD simulation spanning from the upper convection zone to the corona. In the simulation, a localised flux emergence event drives the formation of a (curved) current sheet in the chromosphere, within which a series of plasmoids is formed (Fig. 48c). On the basis of synthesised spectral lines from the simulation the authors suggest diagnostics for plasmoid-mediated reconnection in observations. Peter et al. (2019) have developed a dedicated simulation for understanding UV bursts that begins from a 2D null dome topology, with a shear flow applied on the lower boundary leading to the collapse of the X-point and eventual formation of plasmoids in the current layer (Fig. 48a, b). They have explored different atmospheric stratifications, corresponding to different values of the plasma-\(\beta \) at the reconnection site. It turns out that when \(\beta \ge 1\) the reconnection is not efficient, with only small increases in temperature and slow plasma flows being generated. For \(\beta \ll 1\) an energetic reconnection process is observed, and the authors extrapolate to predict a maximum peak temperature that can be expected from such a process of around 0.2 MK. Plasmoid instability in the low solar atmosphere has also been modelled by Guo et al. (2020), who compare synthetic emissions with IRIS data, and by Ni et al. (2015); Ni and Lukin (2018), who consider the impact on the plasmoid instability of various effects associated with partial ionisation such as ambipolar diffusion and recombination.

Fig. 48
figure 48

Plasmoids in simulations of reconnection in the low solar atmosphere. a, b Temperature in the 2D MHD simulations of reconnection driven by a photospheric flow by Peter et al. (2019). In (a) the peak temperature in the reconnection region as a function of time is plotted, illustrating the bursty nature of the reconnection, while in (b) the temperature distribution is shown at the time illustrated by the blue vertical line in (a), in which two large plasmoids are clearly visible in the current sheet. c Density distribution in the 2.5D radiative MHD simulations of Rouppe van der Voort et al. (2017)

Further observational evidence of bursty, plasmoid-mediated reconnection in small-scale reconnection events on the Sun has been reported by, e.g. Innes et al. (2015), while observed ‘blobs’ in jet/flare/CME current sheets have also been interpreted as plasmoids (e.g., Lin et al. 2005; Takasao et al. 2011; Kumar et al. 2019). These have been complemented by 3D simulations of jets that exhibit tearing and plasmoids in a current sheet formed about a coronal 3D null point (Moreno-Insertis and Galsgaard 2013; Wyper et al. 2016). 2D and 2.5D models of plasmoids in flare/CME current sheets include those by Bárta et al. (2011), Karpen et al. (2012), Lynch et al. (2016) and Hosteaux et al. (2018). Finally, formation of 3D flux ropes during impulsive, bursty reconnection in a current sheet in the laboratory has been reported at the MRX (Magnetic Reconnection eXperiment) by Dorfman et al. (2013).

To summarise, while there remain aspects of nonlinear MHD tearing that are poorly understood—especially in 3D—it provides a viable route to fast reconnection, and one that is expected to be relevant to energy release in many solar applications (Ji and Daughton 2011). It is also possible that MHD tearing creates a hierarchy of current sheets that eventually accesses kinetic scales, at which point additional physics beyond MHD is required to understand the details of reconnection at these smallest scales (Sect. 9). While not the focus of this article, it is notable that fast reconnection mediated by the plasmoid instability has been explored using Hall MHD (Shepherd and Cassak 2010) and full particle-in-cell (e.g., Daughton et al. 2009b) approaches—see Sect. 9.2(ii)(c).

8.4 Response to a resistivity enhancement

Ugai has undertaken many experiments on the fast-reconnection response to a local resistivity enhancement, both in 2D and 3D. These have been performed in geomagnetic tail-like equilibria and in solar coronal arcade equilibria in order to model both geomagnetic substorms and solar flares. In a 2D model, Ugai (1999) allows the resistivity enhancement to depend on temperature. In an initial phase the reconnection grows slowly, but, after plasma and flux have been ejected from near the null point, this is followed by an explosive phase of much faster reconnection due to a positive feedback between the resistivity enhancement and the reconnection flow. The explosive phase sets up fast reconnection at the maximum reconnection rate, with standing slow shocks attached to a localized diffusion region and extending outwards in time. Ugai (2000) extended this analysis to the case in which the fields on both sides of the current sheet are different, finding the growth of an asymmetric plasmoid, predominantly in the region containing the weaker magnetic field. Ugai and Kondoh (2001) consider the effect of resistivity onset threshold (\(V_{c0}\)) and plasma beta (\(\beta \)). When \(V_{c0}\) is large enough, fast reconnection is set up, but when it is too small the diffusion region lengthens and reconnection becomes less effective. Furthermore, it is only when \(\beta \) is small enough that fast reconnection is set up.

Ugai and Wang (1998) and Ugai et al. (2004) extended the spontaneous fast reconnection model to three dimensions. Symmetry conditions were assumed on the xy, yz and zx planes and free boundary conditions on the other boundaries of the first quadrant. Qualitatively, the results are similar to those in 2D. 3D fast reconnection evolves explosively as a nonlinear instability due to a positive feedback between the anomalous resistivity and the reconnection flow. Slow shocks stand in the flow and ahead of the fast reconnection jet. A large-scale 3D plasmoid swells and propagates in the central current sheet, while a vortex flow is formed near the plasmoid side boundary.

Ugai and Zheng (2005) continued their 3D reconnection study and found that fast reconnection does not occur with classical resistivity, which decreases with temperature like \(T^{3/2}\), but it does occur with an anomalous resistivity that increases with current when a threshold is exceeded. Here, resistivity is enhanced in the shock layer which thickens so that secondary tearing is more likely. When the anomalous resistivity increases with the electron-ion drift velocity, fast reconnection evolves rapidly and is sustained steadily. Ugai (2008) applied these ideas to a 3D model of a two-ribbon solar flare, in which the down-flowing fast reconnection jet causes impulsive chromospheric heating by a factor of 30 in two thin layers (or current wedges) near the separatrices, which move apart in time (Fig. 49). This is accompanied by chromospheric evaporation and expanding coronal loops.

Fig. 49
figure 49

Image reproduced with permission from Ugai (2008), copyright by AIP

a Plasma flow vectors and ohmic heating in the \(x=0\), \(y=0\) and \(z=0\) planes for fast 3D reconnection in a coronal arcade initiated by a local enhancement of anomalous resistivity. \(x=0\) represents the chromosphere in which the ohmic heating indicates the site of a flare ribbon, while \(z=0\) shows a vertical cut through the coronal arcade. Distances are normalised with respect to the half-width of the initial current sheet and time with respect to the corresponding Alfvén travel time. b The corresponding magnetic field lines in the \(z=0\) plane and the two main current channels, including the current wedge (CW) that flows to the chromospheric ribbons and a second current channel (\(C_{1}\)) that flows from the fast shock downstream of the reconnection jet.

9 Fast reconnection in a collisional or collisionless medium

Although this review deals mainly with a resistive collisional plasma, there have been many important discoveries recently concerning fast collisionless reconnection, and so we give here a brief overview of fast reconnection, in both a resistive plasma and a collisionless plasma. Apart from the plasmoid instability, we focus on steady-state reconnection. The chapter begins with some general principles for categorising reconnection (Sect. 9.1), which show that reconnection in two dimensions is much more diverse than often realised, so that general statements about reconnection need to have their domain of validity made clear. Then a summary of the results is presented (Sect. 9.2), mainly for ‘almost-uniform’ or local reconnection. Finally, the many unanswered questions that remain for two-dimensional reconnection are described (Sect. 9.3). For reviews of this subject, we have found Cassak et al. (2017) especially insightful, but see also Priest and Forbes (2000); Bhattacharjee (2004); Birn and Priest (2007); Daughton et al. (2009a); Yamada et al. (2010); Ji and Daughton (2011); Daughton and Roytershteyn (2012); Priest (2014); Comisso and Bhattacharjee (2016)

The conclusion detailed below is that three scenarios have been proposed for fast reconnection at a hundredth or a tenth of the Alfvén speed (\(v_A\)), each of which is likely to occur in the solar atmosphere under different circumstances, namely,

  1. (a)

    Petschek reconnection when the plasma is collisional and the resistivity enhanced,

  2. (b)

    Collisionless Hall reconnection when the current sheet width is of order the ion inertial length (\(d^{(i)}\)),

  3. (c)

    Impulsive bursty or plasmoid reconnection when the current sheet goes unstable to tearing.

The apparent existence of a universal rate of reconnection of \((0.01-0.1)v_A\) has long been a puzzle, but we suggest that it arises because reconnection is usually dominated by the ideal MHD processes taking place in the region around the central current sheet (called the diffusion region) and depends only weakly on the microphysics of the current sheet and its length (Priest et al. 2021).

9.1 Principles for categorising fast reconnection

The following distinctions are important for classifying magnetic reconnection. They involve definitions which are common in a traditional solar MHD context, but differ from some definitions that have been adopted more recently in the kinetic plasma community.

  1. (i)

    Local or global reconnection rate

    Most of the plasma behaves in an ideal way, satisfying the ideal equations of MHD, regardless of whether the plasma is resistive or collisionless. In reconnection, one therefore considers an ideal region that surrounds a diffusion region, in which ideal MHD breaks down and the magnetic fields can slip through the plasma. In general, it is important to consider the physics of what is happening in both the ideal region and the diffusion region.

    Regardless of whether the reconnection is resistive or collisionless, the first major point to make is that there is a distinction between the local (or small-scale or microscale) reconnection rate, namely, the speed \(v_i\) with which the plasma carries magnetic flux int