# Magnetohydrodynamic Reconnection

- D. I. PontinD. I. PontinSchool of Mathematical and Physical Sciences, University of Newcastle, Australia

### Summary

Magnetic reconnection is a fundamental process that is important for the dynamical evolution of highly conducting plasmas throughout the Universe. In such highly conducting plasmas the *magnetic topology* is preserved as the plasma evolves, an idea encapsulated by Alfvén’s frozen flux theorem. In this context, “magnetic topology” is defined by the connectivity and linkage of magnetic field lines (streamlines of the magnetic induction) within the domain of interest, together with the connectivity of field lines between points on the domain boundary. The conservation of magnetic topology therefore implies that magnetic field lines cannot break or merge, but evolve only according to smooth deformations. In any real plasma the conductivity is finite, so that the magnetic topology is not preserved everywhere: magnetic reconnection is the process by which the field lines break and recombine, permitting a reconfiguration of the magnetic field. Due to the high conductivity, reconnection may occur only in small *dissipation regions* where the electric current density reaches extreme values. In many applications of interest, the change of magnetic topology facilitates a rapid conversion of stored magnetic energy into plasma thermal energy, bulk-kinetic energy, and energy of non-thermally accelerated particles. This energy conversion is associated with dynamic phenomena in plasmas throughout the Universe. Examples include flares and other energetic phenomena in the atmosphere of stars including the Sun, substorms in planetary magnetospheres, and disruptions that limit the magnetic confinement time of plasma in nuclear fusion devices. One of the major challenges in understanding reconnection is the extreme separation between the global system scale and the scale of the *dissipation region* within which the reconnection process itself takes place. Current understanding of reconnection has developed through mathematical and computational modeling as well as dedicated experiments in both the laboratory and space. Magnetohydrodynamic (MHD) reconnection is studied in the framework of magnetohydrodynamics, which is used to study plasmas (and liquid metals) in the continuum approximation.

### Keywords

### Subjects

- Astronomy and Astrophysics
- Fluid Mechanics
- Plasma Physics

### Introduction

Magnetic fields play a key role in the dynamics of many plasmas throughout the Universe, from galaxies and stars to planetary magnetospheres and laboratory fusion devices. The plasma in many of these environments is highly conducting, in the sense that the magnetic Reynolds number ${R}_{m}=LV\text{/}\eta \gg \text{1}$, where *L* and *V* are typical length and velocity scales, respectively, and *η* is the magnetic diffusivity (*η* = 1/(*μ*_{0}*σ*) where *μ*_{0} is the magnetic permeability and *σ* is the electrical conductivity). In such highly conducting plasmas, magnetic reconnection is a fundamental process, and is associated with rapid energy conversion, driving many dynamic phenomena.

To understand why this is the case, it is instructive to consider the evolution of magnetic flux in a perfectly conducting plasma. In the limit of infinite conductivity—also known as the “ideal” limit—the magnetic induction B (referred to commonly, and hereafter, as the magnetic field) evolves according to the equation

$v$ being the plasma velocity. One consequence of this equation is that

where *S* is any surface that is comoving with the plasma (e.g., Priest, 2014), and *d*/*dt* denotes the “material derivative” or “convective derivative.” The magnetic flux through any comoving surface is therefore preserved. Furthermore, using standard vector identities, Equation (1) can be recast in the form

and using the continuity equation $d\rho \text{/}dt=\u2013\rho \nabla \cdot v$

to eliminate $\nabla \cdot v$ leads to

This is identical to the evolution equation for a Lagrangian line element, *δ***x**, connecting two fluid elements in the flow (e.g., Moffatt, 1978), with *δ***x** replaced by $B/\rho $. Since magnetic field lines are by definition everywhere tangent to $B$, this implies that magnetic field lines are “frozen into” the flow, with the consequence being that all infinitesimal plasma elements connected by a magnetic field line at a given time will remain connected by a field line at all later times. This property is known as “field-line conservation.” Since the plasma velocity $v$ is smooth, it implies that the field lines cannot break or merge as the evolution proceeds. Thus the *magnetic topology*—the connectivity and linkage of magnetic field lines within the domain of interest and between points on the domain boundary—is preserved.

*Note*. The shaded region marks the dissipation region/current layer, blue arrows show plasma flow, and black lines magnetic field lines. Dot-dashed lines in (b) are standing slow-mode shocks.

This conservation of the magnetic topology in an ideal (perfectly conducting) plasma has profound implications for the plasma evolution. In particular, the magnetic field can be stressed by motions that increase the magnetic energy by compressing, stretching, or tangling the field lines. This can lead to situations in which topologically bound excess magnetic energy builds up, that is, magnetic energy that cannot be released without a change of magnetic topology. Of course, in any real plasma the conductivity is finite, and the induction equation becomes

Where $\text{j}=\nabla \times B\text{/}{\mu}_{0}$ is the electric current density. When *σ* is large (so that the magnetic diffusivity *η* = 1/(*μ*_{0}*σ*) is small), magnetic flux and field-line conservation hold (to a good approximation) in the vast majority of the volume; this conservation only breaks down in narrow layers in which intense concentrations of the electric current density develop. In these *current layers*, dissipation becomes significant, the field lines are no longer tied to the plasma and magnetic reconnection may occur. This allows conversion of the stored magnetic energy into bulk kinetic energy, thermal energy, and kinetic energy of non-thermally accelerated particles. As such, reconnection is invoked to explain such diverse phenomena as solar and stellar flares, the unexpectedly high temperature of the Sun’s atmosphere, substorms in planetary magnetospheres, jet acceleration in active galactic nuclei and gamma-ray bursts, angular-momentum transport in accretion disks, and disruptions such as sawtooth crashes in laboratory nuclear-fusion devices.

This article describes both modeling and observations that give insight into the reconnection process. However, it is worth noting that in many applications the magnetic field is difficult to measure with sufficient accuracy to allow reconstruction of the field topology. As such, typically it is the products of reconnection—the kinetic and thermal energy deposited into the plasma—that are actually observable. The focus here is on describing magnetic reconnection in the magnetohydrodynamic (MHD) limit: in this limit the plasma is treated as a continuum. This limit is appropriate for modeling the reconnection process, broadly speaking, when the mean-free path of particles in the plasma is much smaller than the length scale of the current layer (dissipation region) within which the reconnection takes place. It furthermore provides a framework for kinetic descriptions.

### Historical Development

#### Early Ideas

Ideas about magnetic reconnection originate in the proposal by Giovanelli (1946) that magnetic X-points (points in a two-dimensional (2D) magnetic field at which $\left|\text{B}\right|=\phantom{\rule{0.2em}{0ex}}0$ and around which the field lines are hyperbolic) may be preferential sites for the particle acceleration observed in solar flares. Then Dungey (1953) demonstrated that the magnetic field in the vicinity of such an X-point is structurally unstable, and in response to small perturbations is susceptible to the formation of intense electric currents in a process now known as “X-point collapse” (see the section “2D Magnetic Nulls”). He suggested that in this process the magnetic field lines can be “broken and rejoined”—the term field-line “reconnection” was coined by Parker (1957).

#### The Sweet–Parker Model

The first quantitative reconnection model was proposed separately by Sweet (1958) and Parker (1957) and describes the merging of anti-parallel magnetic fields within a planar current sheet of width $2l$ and length $2L$. The oppositely directed magnetic fields are uniform with strength ${B}_{i}$ and are pressed together by a flow $\left({v}_{i}\right)$ (Figure 1a). Magnetic field of strength ${B}_{o}$ is ejected from the end of the current sheet at speed ${v}_{o}$, again both quantities being assumed uniform. The model provides an order-of-magnitude estimate for the rate at which magnetic flux can be “processed” in the current sheet. First, conservation of mass through the boundary of the current sheet requires that

(density is assumed constant). Then, assuming that the steady current sheet is maintained by a balance between inwards advection and outwards diffusion,

Finally, assuming dominance of magnetic pressure in the inflow and dynamic pressure in the outflow, then a steady state requires that these balance:

where ${v}_{Ai}$ is the Alfvén speed in the inflow region. Now, eliminating $l$ and ${v}_{o}$ between these three relations yields an estimate for the inflow speed ${v}_{i}$ at which flux enters the current sheet in the model. This is usually written in dimensionless form as

where the Lundquist number, $S$, is the magnetic Reynolds number based on the Alfvén speed, in this case in the inflow region. The ratio ${v}_{i/}{v}_{Ai}$ is known as the *reconnection rate*, in 2D. The Sweet–Parker model is described as modeling “slow” reconnection due to the rather strong dependence of this reconnection rate on the resistivity, $\eta $. In astrophysical plasmas, $\eta $ may be of order 10^{–10} or smaller (assuming that the formula of Spitzer (1962) can be applied), and as such it was argued that the Sweet–Parker mechanism could not account for rapid energy release in solar flares.

#### Petschek’s Model

In order to try to explain the rapid energy release in solar flares, Petschek (1964) proposed that the current layer of the Sweet–Parker model does not have macroscopic length, but instead is localized, terminating at each end in a pair of standing slow-mode shocks, at which the majority of magnetic energy conversion takes place (Figure 1b). The reconnection rate is determined by the rate at which the flow in the external region, ${v}_{e}$, transports flux toward the current layer, normalized as before to the local Alfvén speed. The model permits a range of values, with increases in reconnection rate being linked to the dimensions of the diffusion region and the angle between the pairs of shock waves. The maximum allowable reconnection rate takes the form

where ${S}_{e}={L}_{e}{v}_{Ae}/\eta $ is the Lundquist number based on the global length-scale ${L}_{e}$. In this case, the reconnection rate is seen to exhibit much weaker dependence on $\eta $, and substituting in appropriate numbers, sufficiently rapid energy release to account for solar flares can be obtained. In the literature the mechanism is therefore referred to as describing “fast reconnection.”

#### Biskamp’s Simulations and Further Steady-State Models

Following the presentation of Petschek’s model, it appeared for many years that the problem of rapid energy release by reconnection was solved. However, this consensus was disturbed when computational simulations of reconnection became feasible, led by the study of Biskamp (1986). Biskamp performed numerical simulations with constant *η*, and obtained current sheets that tended to grow to the system scale, leading to slow Sweet–Parker-like reconnection. Around the same time, Priest and Forbes (1986) developed a set of steady-state reconnection solutions that generalized Petschek’s solution, which was subsequently generalized further still (e.g., Priest & Lee, 1990). Subsequent numerical simulations by Yan, Lee, and Priest (1992) confirmed that these generalized solutions—including Petschek-like solutions—can be obtained by applying appropriate boundary conditions, so long as a region of enhanced resistivity is included around the null point. More recently it was shown (e.g., Forbes, Priest, Seaton, & Litvinenko, 2013) that in order to obtain a localized current sheet it is sufficient to include some spatial variation of *η* near the null. Such resistivity gradients could result, for example, from current-driven micro-instabilities. Nevertheless, it remains to be determined which of these reconnection solutions is applicable in real applications—in which the reconnection process is inherently time-dependent and embedded in a global field evolution—due to their sensitivity to the computational boundary conditions and reliance on a resistivity gradient that is imposed.

#### Time-Dependent Reconnection: The Tearing Mode

In the preceding discussion, reconnection solutions that are both steady-state and 2D were considered. Furth, Killeen, and Rosenbluth (1963) published groundbreaking results on the stability of current layers, in the first time-dependent model. They identified three instabilities of a current layer in the presence of a finite, spatially uniform resistivity: the “tearing,” “rippling,” and “gravitational” modes. Of these the most relevant in terms of reconnection is the tearing mode, since it can occur for wavelengths much longer than the current sheet width. This tearing instability in a neutral sheet (plane on which $\left|B\right|=\phantom{\rule{0.2em}{0ex}}0$) leads spontaneously to reconnection of field lines within the current layer to form “magnetic islands” (see Figure 2). Furth et al. (1963) considered a current layer of infinite length and half-width *l* with large ${S}_{l}$, where ${S}_{l}$ is the Lundquist number based on the thickness of the current layer and can be interpreted as the ratio of the diffusion time $\left({\tau}_{d}\right)$ to the Alfvén travel time $\left({\tau}_{A}\right)$ across the layer. The fastest growing mode has wavenumber $k\text{~}{S}_{l}{}^{-1/4}\text{/}a$, with corresponding growth rate $\gamma =1/\left({\tau}_{A}\sqrt{{S}_{l}}\right)$. Since this growth rate has a similar unfavorable scaling with *η* as the Sweet–Parker model, the tearing mode was originally dismissed as an explanation for rapid energy release in flares. However, interest was revived much later when tearing of a current layer of finite length (together with details of the non-linear phase of the evolution) was considered—see the section “Impulsive, Bursty Reconnection Due to Current Layer Instabilities.”

The development of the tearing instability theory was motivated by understanding plasma behavior in laboratory devices. Later, further interest in reconnection was stimulated by additional insights from laboratory plasma experiments. First, Taylor (1974) invoked a turbulent cascade of reconnection processes in which the only quantity preserved is the magnetic helicity

to explain relaxation in a Reversed Field Pinch. Then Kadomtsev (1975) explained sawtooth relaxation in tokamak plasmas using reconnection.

#### Early Theory of Magnetic Reconnection in Three Dimensions

Motivated by a desire to understand the formation of plasmoids in the Earth’s magnetotail, Schindler, Hesse, and Birn (1988) argued that previous definitions of reconnection were too restrictive, and so developed the theory of “General Magnetic Reconnection.” They showed that for three-dimensional (3D) magnetic fields, reconnection can occur in the absence of null points of $B$. They furthermore demonstrated that the presence of a non-vanishing component of the electric field parallel to the magnetic field, denoted ${E}_{\left|\right|}$, plays a crucial role for reconnection. Specifically, they demonstrated that a breakdown in field-line conservation occurs—crucially for pairs of plasma elements that do not enter the diffusion region—when there exists a region of space within which

where *s* denotes arc length along a magnetic field line. (Note that the volume within which ${E}_{\left|\right|}\ne 0$ must be finite for the process to be classified as reconnection, as opposed to diffusion.) They also related ${E}_{\left|\right|}$ to the change of magnetic helicity during the reconnection process. In a follow-up paper, the mathematical theory was developed in terms of Euler potentials for the magnetic field (Hesse & Schindler, 1988). Here it was demonstrated that the rate at which flux is reconnected is given by the maximal value of $\mathrm{\Psi}$ over all field lines that thread through the dissipation region. These two papers form a key part of the foundation for the theory of 3D reconnection, described in more detail in the remainder of this article.

#### Reconnection Experiments and Regimes

Studies of reconnection have advanced since the early models, through increasingly sophisticated mathematical and computational models, but also through the advent of dedicated experiments in the laboratory and space. Early experiments were able to observe the tearing instability (Anderson & Kunkel, 1969) and steady reconnection (Bratenahl & Yeates, 1970), and subsequent experiments sought to measure reconnection in different configurations, at higher spatial and temporal resolution, and in plasma parameter regimes closer to those found in fusion devices and space plasmas (e.g., Brown, 1999; Frank, 1976; Stenzel & Gekelman, 1981; Syrovatskii, 1981; Yamada, Ono, Hayakawa, & Katsurai, 1990). More recently, the Magnetic Reconnection Experiment (MRX) at Princeton Plasma Physics Laboratory (Yamada et al., 1997) has contributed numerous advances to our understanding through experiments on driven reconnection, including one of the first quantitative verifications of the Sweet–Parker model (Ji, Yamada, Hsu, & Kulsrud, 1998), and first direct observations of the impact of the Hall effect on the current layer geometry (Yamada et al., 2006). These experiments are not surveyed further here, but the interested reader is directed to the review by Yamada, Kulsrud, and Ji (2010) for further information on the device configurations, plasma parameters, and results. It is worth noting that the laboratory is not the only location where direct measurements of reconnection are being made. CLUSTER, and more recently the Magnetospheric Multiscale Mission, are sets of spacecraft launched explicitly to study reconnection processes at the Earth’s magnetopause and magnetotail (e.g., Burch et al., 2016).

It is important at this point to emphasize that the observations of reconnection in the magnetosphere, as well as some of the laboratory experiments, are in plasmas that are not strongly collisional, and hence the MHD approximation is not valid. As such, additional terms in Ohm’s law become important in governing the rate at which reconnection proceeds and the geometry of the reconnecting current layer. The plasma regimes in various astrophysical environments and laboratory experiments have been thoroughly categorized by Ji and Daughton (2011), who provide what they describe as a “phase diagram” for reconnection, which quantifies the importance of these non-MHD effects. It is worth emphasizing that while the remainder of this article discusses the MHD limit, much of what is discussed holds true beyond this limit, and that in many cases the external dynamics outside of the current layer are well described by MHD.

### Fundamental Principles of Reconnection

Recall that in the Introduction magnetic reconnection was defined as a change in the topology of the magnetic field. Two magnetic fields are topologically equivalent if and only if one field can be transformed into the other by means of a smooth (continuously differentiable) deformation. Equivalently, some smooth ideal evolution (flow) can transform one field into the other. Such an evolution between topologically equivalent fields preserves all connections of field lines from one boundary to another and also all linkages or knottedness of field lines within the volume. (Here it is assumed that if magnetic field lines cross the domain boundary then the velocity there vanishes—however, the concept can be generalized to allow for non-zero boundary motions by considering a model external field, though this is omitted here for simplicity.) Therefore, the breakdown of field-line conservation used as a definition for general magnetic reconnection by Schindler et al. (1988) is seen to be equivalent to a change of the magnetic topology. Note that in order to be defined as reconnection, this change of topology must be due to a local non-ideal evolution (as opposed to, e.g., a global diffusion). The relationship between magnetic topology and magnetic reconnection is discussed in more detail by Hornig and Schindler (1996).

In this section, the fundamental properties of reconnection are discussed, many of which are found, surprisingly, to be crucially different between 2D and 3D. Consider rewriting Equation (1) as

or, uncurling,

where the plasma velocity $v$ has been replaced by some other vector field, $w$. Note that no reference has been made here to the explicit form of Ohm’s law, and thus the arguments of this section do not rely on the MHD approximation. Using the same arguments as in the Introduction, this describes the ideal evolution of a magnetic field $B$ whose topology is preserved, provided that $w$ is continuous and differentiable. (Strictly speaking, for topology preservation to hold, the right-hand side of Equation (13) takes the form $\lambda B$, where $\lambda $ is some scalar field—Hornig and Schindler (1996).) The quantity $w$ is called a flux-conserving velocity or flux transport velocity. It remains to be determined under which conditions such a flow can be found, that is, under what conditions is topology preserved (or not)?

Consider first a situation in which $E\phantom{\rule{0.2em}{0ex}}\xb7\phantom{\rule{0.2em}{0ex}}B=\phantom{\rule{0.2em}{0ex}}0$ everywhere, an important example being the case of 2D magnetic field and flow. The solution of Equations (13) or (14) are straightforward: since $E\phantom{\rule{0.2em}{0ex}}\xb7\phantom{\rule{0.2em}{0ex}}B=\nabla \mathrm{\Phi}\cdot B=0$, it follows that $\nabla \mathrm{\Phi}=\delta w\times B$ for some $\delta w$. Then (14) is solved for $w$ to give $w=\delta w+E\times B/\left(B\phantom{\rule{0.2em}{0ex}}\xb7\phantom{\rule{0.2em}{0ex}}B\right)$. Therefore, $w$ always exists, but may be singular at magnetic null points $\left(B=\phantom{\rule{0.2em}{0ex}}0\right)\text{if}\phantom{\rule{0.2em}{0ex}}E\ne 0$ there. By contrast, in 3D the presence of a spatially localized region within which $E\phantom{\rule{0.2em}{0ex}}\xb7\phantom{\rule{0.2em}{0ex}}B\ne 0$—the requirement for 3D reconnection (see the section “Early Theory of Magnetic Reconnection in Three Dimensions”)—provides a sufficient (but not necessary) condition for the non-existence of a continuous, differentiable flow $w$ (Priest, Hornig, & Pontin, 2003).

The singularity of $w$ at 2D magnetic X-points is a signature of the cutting and rejoining of field lines at the X-point. Each plasma element experiences a discontinuous change in the plasma elements to which it is magnetically connected at the instant that it crosses the separatrix of the X-point (separatrix field lines form the boundary between regions of distinct magnetic field-line connectivity—at a 2D “X-point” four separatrix field lines meet at the null forming an “X” geometry). Since $w$ is otherwise continuous and differentiable everywhere, the magnetic connectivity is preserved at all other points in the domain. The reconnection of magnetic field lines in 2D therefore occurs at a single point, in a one-to-one, pairwise fashion. However, 3D reconnection does not share any of these properties with the 2D case (Hesse & Schindler, 1988; Priest et al., 2003). For 3D reconnection in a localized non-ideal region, the magnetic flux evolution exhibits the following properties:

A flux-transporting velocity $w$ *does not* exist for the flux threading the diffusion region (region in which $E\phantom{\rule{0.2em}{0ex}}\xb7\phantom{\rule{0.2em}{0ex}}B\ne 0$).

Therefore, at each instant, *every* field line threading the non-ideal region changes its connectivity. Magnetic field lines that are traced at each instant in time from footpoints comoving in the ideal flow appear to split *as soon as they enter the non-ideal region*, and their connectivity changes *continually and continuously* as they pass through the non-ideal region (see Figure 3).

A natural consequence is that magnetic field lines are *not* reconnected in a one-to-one fashion. In particular, if prior to reconnection a pair of field lines connect plasma elements A to B and C to D, then if A connects to C after reconnection, B will *not* be connected to D (or vice versa).

These properties are demonstrated in Figure 3, which is based on the steady-state kinematic solution for reconnection in the absence of nulls by Hornig and Priest (2003). However, they hold regardless of the structure of the magnetic field in the vicinity of the reconnection region (for example, the presence of nulls and separators, see the following sections). In the figure, representative flux tubes are traced from four cross-sections (black) comoving in the ideal flow, chosen such that at the initial time they coincide pair-wise. As soon as these field lines enter the non-ideal region, the flux tubes from cross-sections A and B (say) are seen to no longer coincide. The apparent “flipping” of field lines (Priest & Forbes, 1992) is shown in Figures 3b–3e, where the opaque sections of the flux tubes move at the local plasma velocity (outside the non-ideal region), while the transparent sections correspond to field lines traced into and beyond the non-ideal region, and appear to flip past one another at a velocity that is different from the local plasma velocity (until the field lines entirely exit the non-ideal region). In Figure 3f it is observed that after reconnection the four cross-sections do not match up to form two unique flux tubes (in contrast to the 2D case).

### Locations/Mechanisms of Current Sheet Formation

Recall that the plasmas under consideration here are high-*R _{m}* plasmas (in which

*η* = 1/(

*μ*

_{0}

*σ*) is very small). Then, considering the theory developed in the section Fundamental Principles of Reconnection and examining Equation (1), we see that a necessary condition for magnetic reconnection is the formation of an intense electric current concentration. Due to the extreme values of

*R*in astrophysical plasmas, one common approach to investigating where such intense current concentrations can develop is to consider the perfectly conducting limit and to seek situations in which the current develops singularities. It can then be argued that—irrespective of how large

_{m}*R*is—the dissipation must eventually become significant as the singular state is approached.

_{m}*Source*: Based on the Solution of Hornig and Priest (2003), after Pontin (2011).

*Note*. Blue and gold are representative magnetic flux tubes in the magnetic field

**B**= (

*y*,

*k*

^{2}

*x*, 1), with

*k*= 1.2. They are traced from footpoints (marked black) that are comoving in the ideal flow. The solid sections of these flux tubes move at the local plasma velocity (outside the non-ideal region), while the transparent sections represent field lines that pass through the non-ideal region and move at a “flipping/slipping” velocity different from the local plasma velocity. The shaded surface in the first frame shows the localized non-ideal region.

#### 2D Magnetic Nulls

As discussed in the section Fundamental Principles of Reconnection, reconnection in 2D occurs at X-type magnetic null points, where the flux velocity may be singular (the same is true at O-type nulls, but this corresponds to flux creation/annihilation). However, in highly conducting plasmas it is still necessary to develop an intense current layer at the null to initiate reconnection. A number of approaches have been made to demonstrating such current accumulation at X-points, and it is instructive to consider them in turn prior to addressing the richer situation found in 3D.

*Linear collapse studies*. As mentioned in the section “Historical Development,” Dungey (1953) was the first to consider the “collapse” of magnetic X-points, studying in particular the region in the close vicinity of the null in which the magnetic field increases linearly with distance from the null. Various authors have extended this analysis, notably in the exact self-similar solution of Imshennik and Syrovatsky (1967), where a finite-time blow-up of the current density is obtained in the ideal limit. The limitation of all such linear models is that the energy that drives the collapse comes from the external region that is not treated in the model.

*Dynamic current growth due to MHD wave accumulation*. Due to the variation in wave speed around X-points, MHD waves tend to be refracted toward the null, with a fraction of the wave energy accumulating at the null (see McLaughlin, Hood, & de Moortel, 2011, and references therein). This drives current growth at the null, distributed in a layer when the wave behaves non-linearly (McClymont & Craig, 1996).

*Sequences of equilibria*. Authors such as Green (1965) and Syrovatskii (1971) argued along the following lines. Consider a 2D potential magnetic field in a closed domain containing an X-point. Now imagine a time evolution of the boundary condition (on, say, $B\cdot \widehat{n}$); one can solve for a sequence of (unique) potential fields consistent with the boundary evolution. However, it may not be possible to connect these through an ideal MHD evolution, in particular because the electric field cannot change at the null point in ideal MHD (since Ohm’s law reads $E+v\times B=0$). Thus the only potential field consistent with the boundary conditions and an ideal evolution must be non-smooth (due to the uniqueness theorem for potential fields). This non-smoothness manifests itself as cuts in the plane, corresponding to singular current sheets.

*Numerical ideal relaxation simulations*. The same approach can be addressed computationally: begin with an equilibrium, apply a perturbation that moves the field lines on the boundaries (specifically the separatrix field lines), and then relax toward a new equilibrium while inhibiting reconnection and holding the boundaries fixed. Using different numerical methods Craig and Litvinenko (2005) and Fuentes-Fernández, Parnell, and Hood (2011) demonstrated that the system indeed develops a current layer that exhibits singular characteristics in the ideal limit.

*Source*: After Pontin (2011).

#### 3D Magnetic Nulls

In 3D, the magnetic field is generically only zero at isolated points: null points. The magnetic field lines in the vicinity of these nulls are hyperbolae that asymptote toward a one-dimensional (1D) *spine* line and a 2D *fan* separatrix surface at large distances (see Figure 4a). These null points are possible sites for the formation of intense currents, as can be shown using analogous approaches to those for X-points in 2D.

*Kinematic arguments*. Consider the mapping between boundary points for field lines in a bounded domain containing an isolated 3D null, as in Figure 4a. Both the spine and fan constitute discontinuities in this field-line mapping: for example, field lines infinitesimally close to one another on either side of the fan map to opposite spine footpoints, and vice versa. As such, in ideal kinematic solutions (in which the induction equation is solved for **v** given an imposed magnetic field), singular velocities are obtained if there is a plasma motion across the spine or fan (Lau & Finn, 1990; Priest & Titov, 1996).

*Dynamic current growth due to MHD wave accumulation*. As in 2D, MHD waves tend to refract toward 3D nulls, with some fraction of the wave energy being trapped around the null, and the current growing until dissipation becomes important (see McLaughlin et al., 2011, and references therein). Alfvén waves deposit their energy along the spine and fan, whereas fast waves refract toward the null itself.

*Numerical ideal relaxation simulations*. Again as in 2D, ideal relaxation simulations for domains containing 3D nulls show that, if the perturbation disturbs the field-line boundary footpoints of the spine or fan from their equilibrium positions, the lowest energy state tends to contain a current layer at the null—the growth of the current in this layer being unbounded in the ideal limit (Fuentes-Fernández & Parnell, 2012, 2013; Pontin & Craig, 2005).

#### Separator Lines

A separator field line is formed by the intersection of two separatrix surfaces (Priest & Titov, 1996), as shown in Figure 4b. By extension of the arguments for 3D nulls in the section “3D Magnetic Nulls,” these can also be shown to be preferential sites for current accumulation. In particular, they also constitute discontinuities of the field-line mapping (Lau & Finn, 1990). Moreover, by considering the topological admissibility of sequences of equilibria, Longcope and Cowley (1996) argued that equilibria containing tangential discontinuities at the separator should result from certain perturbations, similar to the approach for 2D X-points by Syrovatskii (1971).

#### Quasi-Separatrix Layers

At null points, separatrices, and separators, the field-line mapping is discontinuous. It was also suggested—and later observed in numerical simulations—that currents tend to accumulate in narrow layer-like regions called *quasi-separatrix layers* (QSLs) where the field-line mapping exhibits large gradients (Démoulin, Henoux, Priest, & Mandrini, 1996; Priest & Démoulin, 1995). These gradients are quantified by the squashing factor of the field-line mapping (Titov, 2007; Titov, Hornig, & Démoulin, 2002), denoted *Q*. *Q* is large in QSLs, and true separatrices can be thought of as a limiting case of a QSL (Démoulin, 2006). QSLs are demonstrated to be preferential sites of current growth when subjected to appropriate perturbations (Effenberger, Thust, Arnold, Grauer, & Dreher, 2011; Galsgaard, Titov, & Neukirch, 2003; Titov, Galsgaard, & Neukirch, 2003). The precise link between *Q*, the perturbation, and the current growth remains to be established, but even in the perfectly conducting limit the current layer is expected to have finite width and intensity.

#### Ideal Instabilities

Formation of current sheets can also be triggered by an ideal instability. For example, sufficiently twisted magnetic flux tubes are susceptible to an ideal *kink instability* which causes a loss of cylindrical symmetry, leading to the formation of a helical current sheet (Hood & Priest, 1979; Kruskal, Johnson, Gottlieb, & Goldman, 1958). It has been known for some time that when the kinking flux tube is periodic—or equivalently indefinitely long—the non-linear evolution of the instability leads to a singular current sheet in the ideal limit. However, in a geometry relevant to, say, the solar atmosphere, the flux tube would be “anchored” at both ends in the dense plasma of the solar interior. This is modeled by *line-tying*; essentially locations of field-line footpoints are fixed at some surface (usually the domain boundary), equivalent to fixing $B\cdot \widehat{n}$. Line-tying is thought to smooth out the current sheet into a current layer of finite width and intensity, increasingly so for shorter flux tubes (see Huang, Bhattacharjee, & Zweibel, 2010, and references therein).

Another classical ideal instability that yields singular current sheets in the ideal limit in the absence of line-tying is the coalescence instability between current-carrying flux tubes (Finn & Kaw, 1977). In 2D, the singular current sheet forms at an X-point. However, when a magnetic-field component is present along the tubes and the effect of line-tying is included, then a QSL exists rather than an X-point, and both the width of the current layer obtained and the maximum current density within the layer are finite (Longcope & Strauss, 1994).

#### Magnetic Braids

Finally, another proposed mechanism for current sheet formation was put forward as part of a mechanism to explain the heating of the Sun’s atmosphere by Parker (1972), which relates to magnetic fields in which all field lines connect between two perfectly conducting planes (see Figure 5). Parker’s hypothesis states that for such a field in which the field lines are sufficiently tangled, no corresponding smooth equilibrium exists. It is then proposed that an ideal relaxation leads toward a magnetic field containing tangential discontinuities—singular current sheets. The hypothesis has attracted significant debate, with a number of arguments—either mathematical or computational—being put forward both supporting (e.g., Janse, Low, & Parker, 2010; Low, 2006; Ng & Bhattacharjee, 1998) and opposing (Craig & Sneyd, 2005; Longcope & Strauss, 1994; van Ballegooijen, 1985; Wilmot-Smith, Hornig, & Pontin, 2009; Zweibel & Li, 1987) this “spontaneous” current-sheet formation. What is now clear is that non-singular equilibria exist for at least some highly tangled or braided magnetic fields, but that as the field complexity increases (as quantified by measures of the field-line mapping) the equilibria must contain current layers that are increasingly thin and intense (Pontin & Hornig, 2015).

### Magnetic-Reconnection Regimes

In this section, the characteristics of single, isolated reconnection processes are described, occurring at the various magnetic structures known to be sites of current-sheet formation as described in section “Locations/Mechanisms of Current Sheet Formation.”

#### 2D X-Point Reconnection

As discussed in the section “Fundamental Principles of Reconnection,” reconnection in 2D occurs at X-points, where field lines reconnect in a one-to-one fashion at the moment that they cross the separatrices. The rate at which flux is transferred across the separatrices is measured by the value of **E** at the X-point. In principle this type of reconnection can also occur in a real 3D magnetic field, although exact invariance in 1D in a finite 3D volume is an unstable state, and in the generic case this symmetry is broken, leading to a fully 3D magnetic field. The types of reconnection that may occur in the real 3D case are much more varied, and in the following sections the different modes of 3D reconnection are discussed.

As discussed in the section “Historical Development,” a significant part of reconnection research has focused on understanding what determines the reconnection rate. In 2D this is unambiguously defined as the rate at which magnetic flux is transported across the separatrices of the X-point. In MHD the reconnection rate (determined to a large extent by the current-layer geometry) is highly sensitive to the boundary conditions and/or presence of variations in the resistivity (see the section “Biskamp’s Simulations and Further Steady-State Models”). In a focused study, Birn et al. (2001) demonstrated that inclusion of additional effects in Ohm’s law beyond single-fluid MHD could in many cases inhibit the length-wise growth of the current layer, leading to enhanced reconnection rates. However, it was subsequently shown that these results were strongly influenced by the periodic boundary conditions employed in the current-sheet’s outflow direction, and that fast reconnection can occur in the MHD regime with uniform resistivity, in response to a current-layer instability (see the section “Impulsive, Bursty Reconnection Due to Current Layer Instabilities”). How the reconnection rate, and important physics in the current layer, are dependent on the external conditions, including whether the reconnection is “forced” as in Taylor’s model (see Wang & Bhattacharjee, 1992, and references therein) or occurs spontaneously (without any conditions imposed on the inflow to the reconnection site), remains an outstanding problem. Reviews of the reconnection rate problem can be found in Comisso and Bhattacharjee (2016) and Cassak, Liu, and Shay (2017).

#### Reconnection at 3D Null Points

As discussed in the section “3D Magnetic Nulls,” the field in the vicinity of 3D magnetic nulls is prone to collapse, generating an intense current layer. Depending on the characteristics of the perturbations that drive this collapse, and thus the characteristics of the current layer, different modes of null-point reconnection have been shown to occur, as categorized by Priest and Pontin (2009). The most common of these is spine-fan reconnection, which occurs when the spine and fan collapse locally toward one another, generating an electric current layer in the fan surface within which the current vector is parallel to the fan surface (see Figure 6). The field evolution involves a transfer of magnetic flux through/past the spine line and through the fan separatrix surface. This is in some sense a combination of the early “spine” and “fan” modes described by Priest and Titov (1996), which it later turned out could not be decoupled in a dynamic evolution, although pure fan reconnection is dynamically accessible in an incompressible plasma (Craig & Fabling, 1996; Pontin, Bhattacharjee, & Galsgaard, 2007b). The maximization of the integral in Equation (12) to obtain the reconnection rate is found to be obtained along field lines in the fan surface, and measures the rate at which flux is transported across the fan (Pontin, Hornig, & Priest, 2005). This mode of null-point reconnection has been identified in laboratory experiments (Frank, 1999; Gray, Lukin, Brown, & Cothran, 2010), implicated in energetic events in the Sun’s atmosphere (Masson, Pariat, Aulanier, & Schrijver, 2009; Pariat, Antiochos, & DeVore, 2009; Yang, Guo, & Ding, 2015), and studied in detail in numerous computer simulations (e.g., Galsgaard & Pontin, 2011; Pontin, Bhattacharjee, & Galsgaard, 2007a).

It turns out that a completely different type of reconnection may take place at 3D null points in addition to the spine-fan mode. In particular, certain types of driving may lead to the accumulation of field-line twist at either the null spine or fan (Fuentes-Fernández & Parnell, 2012; Rickard & Titov, 1996). In such a configuration, the electric-current vector at the null point itself is oriented parallel to the spine line, leading to a rotational slippage of field lines within the current layer (Pontin, Hornig, & Priest, 2004), termed torsional spine or torsional fan reconnection by Priest and Pontin (2009), depending on the localization of the current (see Figure 6). Note that while this reconnection mode relieves accumulated stress, it does not involve flux transfer between topologically distinct flux domains.

#### Reconnection at Separators

Reconnection at separator field lines is perhaps the most natural extension from the 2D reconnection picture, since a separator is defined by the intersection of two separatrix surfaces (as is an X-point in 2D). Early kinematic models in current-free magnetic fields involved a simple cut-and-paste of field-line pairs at the separator line (Lau & Finn, 1990; Priest & Titov, 1996) so that the process appeared much like 2D X-point reconnection when projected on such a 2D plane perpendicular to the separator (see Figure 7). However, as demonstrated by Parnell, Haynes, and Galsgaard (2010), the structure of the magnetic field in the vicinity of the separator may be significantly more complex than previously expected, changing from hyperbolic (“X-type”) to elliptic (“O-type”) at different locations along the separator. What is clear is that the reconnection must occur in a finite volume around the separator line, consistent with the properties described in the section “Fundamental Principles of Reconnection.” The detailed flux evolution and reconnection flows have been studied in a number of simulations (Parnell et al., 2010; Stevenson & Parnell, 2015a, 2015b), but a general picture remains to be determined, as does the role of the null points (from which emanate the separatrix surfaces that form the separator). Separator reconnection was identified during the reconfiguration of the coronal magnetic field in observations of magnetic flux emergence on the Sun (Longcope, McKenzie, Cirtain, & Scott, 2005). It is thought to be important, in particular, at the dayside of the Earth’s magnetosphere (see Glocer, Dorelli, Toth, Komar, & Cassak, 2016, and references therein).

*Source*: Modified from Pontin (2011).

*Note*. (a) Schematic of the structure around a null point during the processes of spine-fan reconnection. Grey and black lines are magnetic field lines that show the local field collapse. Black arrows indicate the plasma flow. The shaded grey surface outlines the current layer, while the red arrows within it indicate the dominant current-vector orientation. (b, c) Schematics showing (b) torsional-spine and (c) torsional-fan reconnection, using the same conventions as in (a).

#### 3D Reconnection in the Absence of Null Points

As described in the section “Locations/Mechanisms of Current Sheet Formation,” the accumulation of intense electric currents in 3D does not require the presence of nulls/separators. This leads to the notion of “non-null reconnection” or “$B\ne 0$ reconnection.” The essential properties of $B\ne 0$ reconnection in a fully 3D configuration were described by Hornig and Priest (2003). Note that here “fully 3D” means that there exists a finite volume within which $E\xb7B\ne 0$, and this is localized in all three directions (cf. Equation (12)). Solving the kinematic, resistive MHD equations, Hornig and Priest (2003) showed that counter-rotational plasma flows are required to maintain a steady-state reconnection solution. More precisely, the flow is confined to the volume of space defined by the magnetic flux that threads the diffusion region, *D*, with plasma on either side of *D* (with respect to the direction of $B$) rotating in opposite senses. As a consequence of the counter-rotational flows, field lines followed from the ideal region above and below *D* undergo a “rotational slippage” with respect to one another, which is quantified by the reconnection rate calculated via Equation (12).

It is worth emphasizing that this characteristic flow structure for 3D non-null reconnection is very different to the stagnation-point flow characteristic of the 2D case. As such, the reconnection rate in this model does not have the same straightforward interpretation as the rate of flux transport across a separatrix structure—as it does for reconnection at an X-point, spine-fan 3D null-point reconnection, or separator reconnection.

The solution of Hornig and Priest (2003), however, allows for the addition of an ideal flow, and a stagnation-type flow can be added to transport magnetic flux into and out of the diffusion region. The result is that field lines are brought into the non-ideal region, are split apart by the counter-rotational flows, and exit differently connected in opposite quadrants of the flow, resulting in the apparent “flipping” motion of field lines shown in Figure 3. The field-line flipping/slippage associated with 3D non-null reconnection has subsequently been detected in simulations of the full set of resistive MHD equations (Pontin, Galsgaard, Hornig, & Priest, 2005), observations of the Sun (Aulanier et al., 2007), and in laboratory plasmas (Lawrence & Gekelman, 2009). When the field-line mapping in the domain exhibits strong gradients (say in a QSL), this apparent “flipping” motion of the field lines can be faster than the local Alfvén speed, in which case Aulanier, Pariat, Démoulin, and Devore (2006) argue that the behavior is in some sense similar to that at a true separatrix, and they describe this as “slip-running” reconnection.

### Turbulent Reconnection

The section “Magnetic-Reconnection Regimes” discussed isolated, individual reconnection processes, and indeed the majority of previous theoretical and experimental studies deal with this case. However, it is becoming increasingly clear that for many plasmas—in particular those with very high magnetic Reynolds number—the generic situation is one in which reconnection occurs in numerous different, but interrelated, processes within the domain. In this section, two different situations are considered. In the first, the reconnection sites are spread throughout the domain (see the section “Reconnection in a Fragmented Dissipation Region”), while in the second reconnection a “large-scale” current layer is modified by the presence of small-scale turbulence (see the section “Impulsive, Bursty Reconnection Due to Current Layer Instabilities”).

#### Reconnection in a Fragmented Dissipation Region

Consider first the situation where reconnection occurs in many current layers spread throughout the domain. One classical scenario that conforms to this picture is Parker’s model for coronal heating by magnetic braiding (Parker, 1972; also recall the section “Magnetic Braids”). The model involves magnetic field lines connecting two perfectly conducting planes (that model sections of the photosphere), the boundary footpoints of these field lines being transported by flows that represent convective flows on the Sun. Numerous studies (see the review by Wilmot-Smith, 2015) have simulated this system, either under continuous boundary driving of various types (Galsgaard & Nordlund, 1996; Rappazzo, Velli, Einaudi, & Dahlburg, 2008) or as a relaxation process without driving (Pontin, Wilmot-Smith, Hornig, & Galsgaard, 2011). In each case, the evolution involves the formation of numerous current layers that are spread throughout the domain. The extent to which the turbulence develops depends on the dissipation, while for resistivity *η* below a critical value, at most a weak dependence on *η* is observed of both the energy dissipation (Rappazzo et al., 2008) and global reconnection rate (Pontin et al., 2011; this global reconnection rate being a sum over many individual reconnection sites).

In the braiding scenario, reconnection occurs in the absence of any nulls or separators. Reconnection in a fragmented dissipation region is also observed in the presence of nulls, for example in simulations of two boundary magnetic-flux patches driven past one another in the presence of an overlying field. Parnell, Haynes, and Galsgaard (2008) found in their simulations that reconnection occurred at numerous separator lines, and they identified a “recursive reconnection” of the magnetic flux. More precisely, they found that magnetic flux was reconnected cyclically through the identified reconnection sites (separators). Surprisingly, each unit of magnetic flux was reconnected, on average multiple (~ 2–4) times as the field sought a new equilibrium, a phenomenon also observed in the magnetic-braid relaxation simulations of Pontin et al. (2011).

#### Impulsive, Bursty Reconnection Due to Current Layer Instabilities

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 reconnective processes. As described in the section “Time-Dependent Reconnection,” the tendency of such a monolithic current layer to fragment into a “chain” of current layers separated by 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. However, Priest and Forbes (1986) suggested this as the basis for a new regime of impulsive bursty reconnection, and Loureiro, Schekochihin, and Cowley (2007) showed—by considering a current layer of large (but finite) aspect ratio—that this “plasmoid instability” can grow rapidly, scaling only weakly with the dissipation, specifically as

*S*

^{1/4}(where

*S*is the Lundquist number, see Equation (9)). High-resolution MHD simulations in 2D have since confirmed that, once the current-layer aspect ratio is greater than ~100 (as predicted by Loureiro et al., 2007), the current layer indeed fragments, with the formation of multiple plasmoids, and the reconnection rate becomes nearly independent of the resistivity (Bhattacharjee, Huang, Yang, & Rogers, 2009; Loureiro, Samtaney, Schekochihin, & Uzdensky, 2012). Subsequently, models of fast reconnection by the plasmoid instability have been developed in Hall MHD (MHD in which the Hall effect is included in the induction equation) (Shepherd & Cassak, 2010), full particle-in-cell (PIC) (Daughton et al., 2009), and in 3D (Daughton et al., 2011; Huang & Bhattacharjee, 2016; Wyper & Pontin, 2014). In 3D in the MHD regime, the process of rapid energy release, and transition to turbulence, may be facilitated in addition by a secondary (ideal) kinking instability of flux ropes within the current layer (Dahlburg & Einaudi, 2002; Dahlburg, Klimchuk, & Antiochos, 2005).

It is worth noting that these results are consistent with earlier work by Lazarian and Vishniac (1999), who argued that the presence of (weak) turbulent fluctuations in a reconnecting current layer should render the reconnection rate essentially independent of resistivity. Kowal, Lazarian, Vishniac, and Otmianowska-Mazur (2009) later supported these results using numerical simulations. It was subsequently demonstrated by Eyink, Lazarian, and Vishniac (2011) that the predictions of Lazarian and Vishniac (1999) can also be derived within a framework that considers field-line “wandering” in the presence of MHD turbulence. Using an analogy with Richardson diffusion of Lagrangian particle trajectories, they analyzed the separation of trajectories in a turbulent flow. Their considerations imply that in the ideal limit ($\eta \to 0$), infinitely many field lines pass through any given point in space. Therefore, the concept of following a fluid element in time breaks down, as does the idea of field-line conservation in the flow. As such, the frozen-flux theorem breaks down in a strict, deterministic sense—without having to appeal to resistive diffusion.

### Conclusions and Open Questions

Magnetic reconnection is a fundamental process that is important for the dynamical evolution of highly conducting plasmas throughout the Universe. It permits a change in the topology of the magnetic field in such plasmas, often accompanied by the (rapid) conversion of magnetic energy to plasma kinetic energy (both of the bulk flow and non-thermally accelerated particles) and heat. Early understanding of reconnection was based on 2D MHD models. In the last 25 years or so, the focus has shifted, with the majority of reconnection research focusing on the 3D problem (primarily in the MHD limit) or on plasma kinetic effects beyond the MHD treatment (primarily in 2D). The focus of this article is on reconnection in the MHD limit. Reviews of non-MHD effects in reconnection can be found in Yamada et al. (2010), Hesse, Neukirch, Schindler, Kuznetsova, and Zenitani (2011), and Zweibel and Yamada (2016).

Understanding magnetic reconnection in plasma systems is inherently a multi-scale problem. Reconnection is dynamically relevant in highly conducting plasmas, and as such the process itself takes place in dissipation regions (current layers) whose dimensions are small compared to the system size. The coupling between the dynamics on global and local scales remains one of the biggest challenges in understanding the implications of reconnection processes. This is highly challenging both experimentally and computationally due to the extreme scale separation. Significant questions that remain open—some related to the coupling of local and global scales—include:

Where does reconnection most readily occur in complex 3D magnetic fields?

What controls the reconnection rate? Indeed, how is the reconnection rate best measured and interpreted in the case of a highly fragmented reconnection site with many interrelated energy conversion processes?

What is the link between reconnection and turbulence?

What is the structure of the current layer at astrophysical parameters, and what is the important physics in the current layer?

Why does reconnection exhibit a sudden onset on many systems (the so-called “trigger problem”)? (For rapid conversion of substantial amounts of energy, an initial period of slow accumulation of magnetic energy is required, followed by a sudden transition to fast energy conversion.)

What determines what fraction of the magnetic energy is converted into heat, kinetic energy of bulk motion, and kinetic energy of (non-thermally) accelerated particles? (The details of this energy budget for reconnection, and how universal it is, remain to be understood.)

How is reconnection modified in “extreme astrophysical environments” (for example, in black-hole accretion disks, neutron-star atmospheres, *γ*-ray bursts)? (Examples of additional effects to be explored include special-relativistic effects and pair creation.)

All these questions and more ensure that magnetic reconnection will remain a highly active research topic for the foreseeable future.

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