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Lecture 33
Magnetic Reconnection

Overview

Magnetic reconnection is not merely “field lines changing partners.” It is the release of magnetic stress stored in the outer ideal region, funneled through a very thin non-ideal layer that allows the topology to change. That is why reconnection, like tearing, is intrinsically non-local.

Historical Perspective

Magnetic reconnection entered plasma astrophysics through attempts to explain solar flares. Giovanelli argued in 1946 that flare activity could be powered by magnetic field restructuring, Dungey sharpened the idea by emphasizing localized departures from ideal behavior, and the first quantitative resistive-MHD scaling followed in the work of Parker and Sweet in the late 1950s (Giovanelli, 1946; Dungey, 1953; Parker, 1957; Sweet, 1958). Soon afterward, Syrovatskii emphasized the spontaneous formation of thin current sheets in frozen-in magnetic fields and the Y-point geometry of long current layers (Syrovatskii, 1971, 1981). Petschek then made the central question unavoidable: if a collisional Sweet–Parker layer is long and thin, how can reconnection ever be fast in a plasma with a huge Lundquist number? (Petschek, 1964)

Reconnection is therefore a classic problem for the same reason the tearing mode is a classic problem: a mathematically tiny non-ideal region can release free energy stored on a global scale. The sheet is local. The drive is not. That local–global structure is the real lesson.

33.1 A thought experiment: two current-carrying flux tubes

Why the tubes attract. Consider two neighboring flux tubes carrying currents in the same \(z\) direction. At the crudest level one may idealize them as two line currents \(I\) separated by a distance \(d\). Then the magnetic force per unit length is

\[\frac {F_{\rm att}}{L} = \frac {\mu _0 I^2}{2\pi d}, \tag{33.1}\]
so parallel currents attract. That attraction is not a small detail. It is the first hint that reconnecting systems usually contain outer-region magnetic free energy. If the two tubes can move toward one another while preserving flux, magnetic energy drops.

In a real flux-tube geometry the field is not that of two literal wires, but the same conclusion survives: like-directed currents attract, adjacent flux bundles are pushed together, and a reversal of the reconnecting field develops in the narrow gap between them. This is the quasi-static version of what later appears in linear tearing theory as a positive outer-region matching parameter \(\Delta '\). The thin inner layer does not create the attraction; it is created because the outer ideal field wants the tubes to merge.

The reactive current sheet between them. Suppose the reconnecting field just above and below the midplane \(y=0\) is

\[\vect {B}(y\rightarrow 0^\pm )= \pm B_0\,\hat {\vect {x}},\]
with the current flowing in the \(z\) direction. Ampère’s law for a tangential field jump gives the surface current
\[\vect {K} = \frac {1}{\mu _0}\,\hat {\vect {y}}\times \left [\vect {B}\right ]_{0^-}^{0^+} = -\frac {2B_0}{\mu _0}\,\hat {\vect {z}}. \tag{33.3}\]
So the current induced in the thin sheet is opposite to the current carried by the two flux tubes. If the sheet has thickness \(\delta \), then the volume current density is of order
\[J_z \sim -\frac {2B_0}{\mu _0\delta }. \tag{33.4}\]
This is the enormous, localized current that resists the merger. As the separation shrinks, \(\delta \) shrinks, and the required sheet current becomes very large.

Caution

The current sheet is not the reason the two flux tubes attract. It is the reaction layer that appears because the outer ideal field tries to bring them together. That point matters later: the same outer attraction shows up in tearing theory as \(\Delta '\), and in Sweet–Parker theory as the global field \(B_0\) and sheet length \(L\) that determine the local layer.

Pressure created at the X line. Now ask what holds this compressed layer in transverse force balance. Across the sheet the normal component of the momentum equation reduces to pressure balance,

\[p + \frac {B^2}{2\mu _0} \simeq p_{\rm up} + \frac {B_0^2}{2\mu _0}, \tag{33.5}\]
where \(p_{\rm up}\) and \(B_0\) are the upstream pressure and reconnecting field. At the exact X line, for zero guide field, the magnetic field vanishes, so the pressure at the center of the sheet is
\[p_X = p_{\rm up} + \frac {B_0^2}{2\mu _0}. \tag{33.6}\]
That equation is worth pausing over. The magnetic pressure of the incoming flux is converted into a real plasma pressure in the middle of the sheet. The sheet is hot and overpressured because magnetic stress has nowhere else to go. In a collisional layer this overpressure is built by compression and Ohmic heating; in weaker-collisionality layers tensor pressure and kinetic effects matter, but the basic statement survives.

If a guide field \(B_g\) threads the layer and is nearly the same on both sides, then it simply cancels out of (33.5). It does not disappear physically, but it does not change the leading-order transverse balance set by the reconnecting component.

Why the pressure drop launches the exhaust. Along the symmetry line of the sheet, near the X line, the magnetic field vanishes to leading order, so the outflow equation is initially just

\[\rho v_x\frac {dv_x}{dx} \simeq -\frac {dp}{dx}. \tag{33.7}\]
Integrating from the X line to the exhaust gives the Bernoulli-like estimate
\[\frac {1}{2}\rho v_{\rm out}^2 + p_{\rm out} \simeq p_X. \tag{33.8}\]
Using (33.6),
\[\frac {1}{2}\rho v_{\rm out}^2 \simeq \left (p_{\rm up} + \frac {B_0^2}{2\mu _0}\right )-p_{\rm out}. \tag{33.9}\]
If the downstream pressure is not much larger than the upstream pressure, then
\[v_{\rm out} \sim \frac {B_0}{\sqrt {\mu _0\rho }} \equiv V_A. \tag{33.10}\]
This is the cleanest way to see where the Alfvénic exhaust comes from. Near the X line it is the pressure drop that starts the flow; farther downstream it is equally natural to say that magnetic tension in the bent reconnected field lines continues to accelerate the exhaust. The two descriptions are complementary, not contradictory.

Why reconnection is non-local. For a tearing-parity perturbation in slab geometry, one introduces a flux function \(\psi (x)\) and defines

\[\Delta ' \equiv \left [\frac {\psi '}{\psi }\right ]_{0^-}^{0^+}. \tag{33.11}\]
This quantity is determined entirely by the outer ideal solution; the resistive inner layer only knows about it through matching. For the standard Harris sheet one finds
\[\Delta ' a = 2\left (\frac {1}{ka}-ka\right ), \tag{33.12}\]
so long-wavelength disturbances with \(ka<1\) have \(\Delta '>0\). That positive \(\Delta '\) is the linear version of the flux-tube-attraction thought experiment above: bringing like-current flux systems together lowers outer magnetic energy.

This is why reconnection, like tearing, is not a purely local diffusion problem. The inner layer decides how rapidly topology changes. The outer ideal field decides whether there is any energy available to release and what macroscopic length \(L\) the layer inherits.

33.2 Sweet–Parker as a local layer controlled by global geometry

Ohm’s law, induction, and the Lundquist number. We now pass from the thought experiment to the textbook steady 2D reconnecting sheet. Let the sheet have length \(L\) and thickness \(\delta \ll L\), with a slow inflow \(v_{\rm in}\) and a fast outflow \(v_{\rm out}\). Using the resistive form of Ohm’s law,

\[\vect {E} + \vect {v}\times \vect {B} = \eta _\Omega \vect {J}, \tag{33.13}\]
and defining the magnetic diffusivity
\[\eta _m \equiv \frac {\eta _\Omega }{\mu _0}, \tag{33.14}\]
the induction equation is
\[\partial _t \vect {B} = \nabla \times (\vect {v}\times \vect {B}) + \eta _m\nabla ^2\vect {B}. \tag{33.15}\]
The Lundquist number based on the global length \(L\) is
\[S \equiv \frac {LV_A}{\eta _m} = \frac {\mu _0 L V_A}{\eta _\Omega }. \tag{33.16}\]

The uniform reconnection electric field. In steady 2D reconnection, Faraday’s law implies that the out-of-plane electric field is spatially uniform. So the same \(E_z\) can be evaluated in the ideal upstream region or in the non-ideal sheet:

\[E_z \simeq v_{\rm in} B_0 \qquad \text {(upstream)}, \qquad E_z \simeq \eta _\Omega J_z \qquad \text {(sheet)}. \tag{33.17}\]
With (33.4), or equivalently \(J_z\sim B_0/(\mu _0\delta )\) up to factors of order unity, this gives
\[v_{\rm in} B_0 \sim \eta _\Omega \frac {B_0}{\mu _0\delta } \quad \Rightarrow \quad v_{\rm in} \sim \frac {\eta _m}{\delta }. \tag{33.18}\]
So inflow can only proceed as fast as flux diffuses across the sheet thickness.

Mass conservation and the outflow speed. For the incompressible textbook estimate,

\[v_{\rm in} L \sim v_{\rm out}\,\delta . \tag{33.19}\]
Using the outflow estimate (33.10),
\[v_{\rm in} \sim V_A\frac {\delta }{L}. \tag{33.20}\]
Combining (33.18) and (33.20) gives
\[V_A\frac {\delta }{L} \sim \frac {\eta _m}{\delta } \quad \Rightarrow \quad \delta ^2 \sim \frac {\eta _m L}{V_A},\]
so that
\[\boxed {\delta \sim L S^{-1/2}} \tag{33.22}\]
and therefore
\[\boxed {v_{\rm in} \sim V_A S^{-1/2}.} \tag{33.23}\]
The dimensionless reconnection rate is then
\[\boxed {\mathcal R \equiv \frac {v_{\rm in}}{V_A} \sim S^{-1/2}.} \tag{33.24}\]

What is really slow about Sweet–Parker. The global reconnection time is

\[\tau _{\rm rec} \sim \frac {L}{v_{\rm in}} \sim \frac {L}{V_A}S^{1/2}. \tag{33.25}\]
So the problem is not that the non-ideal term is tiny in the sheet; that is true in many plasmas. The problem is that a long thin current layer forces a tiny inflow speed. The geometry itself creates the bottleneck.

Note

Sweet–Parker is slow because the same balance that makes the layer thin, \(\delta /L\sim S^{-1/2}\), also makes it long and slender. The diffusion region is too elongated to accept flux rapidly.

A pressure and energy interpretation. The same result can be read as an energy-conversion statement. Magnetic energy enters as Poynting flux of order

\[S_{\rm mag} \sim \frac {E_z B_0}{\mu _0} \sim \frac {v_{\rm in}B_0^2}{\mu _0}. \tag{33.26}\]
That energy does not emerge purely as directed flow. Part goes into the exhaust kinetic energy, part into the elevated pressure (33.6), and part into local dissipation through Joule heating \(\eta _\Omega J^2\). This is why experiments that measure downstream pressure, exhaust throttling, and effective resistivity are so informative: they are testing the same pressure-balance story that the simple scaling already contains.

33.3 X points, Y points, and long current layers

Syrovatskii layers. The Sweet–Parker picture is often drawn as a rectangular box, but real current sheets are not obligated to stay rectangular. Syrovatskii emphasized that a thin current layer can terminate in Y points, so that the reconnecting geometry consists of a long, compressed sheet joined onto the external ideal field through singular endpoints (Syrovatskii, 1971, 1981). That language is useful because it reminds us that the diffusion region cannot be specified independently of the outer field: the sheet and its endpoints are a single matched structure.

Petschek’s objection. Petschek’s key insight was geometrical. If the non-ideal region were short rather than of order \(L\), the severe Sweet–Parker bottleneck might be avoided. His solution replaced the single long resistive sheet by a much shorter inner diffusion region joined to standing slow shocks (Petschek, 1964). Whether a given plasma actually realizes that geometry is a dynamical question, but the point of the objection has survived: reconnection rates are controlled as much by the global shape of the layer as by the microscopic term that breaks ideal MHD.

Solar Y points and helmet streamers. This is more than a historical cartoon. In the solar corona, cusp and Y-point geometries appear naturally above helmet streamers, and the heliospheric current sheet emerges from that global streamer topology. The classical Kopp–Pneuman flare cartoon is therefore valuable not because every solar flare literally looks like the cartoon, but because it makes the long-sheet/Y-point geometry visible (Kopp and Pneuman, 1976). It also provides a clean bridge back to Lecture 8, where the Parker spiral and helmet-streamer geometry were introduced on the solar-wind side. The point is the same in both lectures: large-scale magnetic geometry creates preferred places for current-sheet formation.

33.4 When the Sweet–Parker sheet tears itself apart

The current sheet as a tearing equilibrium. The Sweet–Parker model already contains the seed of its own destruction. Because \(\delta _{\rm SP}/L\sim S^{-1/2}\), the aspect ratio grows like \(L/\delta _{\rm SP}\sim S^{1/2}\). At sufficiently large Lundquist number a single Sweet–Parker sheet is not a robust steady object; it is unstable to tearing and fragments into a chain of plasmoids (Loureiro et al., 2005, 2007; Uzdensky et al., 2010; Bhattacharjee et al., 2009; Huang and Bhattacharjee, 2010).

To connect this directly with the tearing lecture, define a local half-thickness \(a\sim \delta _{\rm SP}\) and local Alfvén time

\[\tau _{A,a}\equiv \frac {a}{V_A}, \qquad S_a \equiv \frac {aV_A}{\eta _m}.\]
Since \(a\sim L S^{-1/2}\),
\[S_a \sim S^{1/2}. \tag{33.28}\]
If \(\kappa \equiv kL\) is the global along-sheet wave number, then
\[ka = \kappa \frac {a}{L} \sim \kappa S^{-1/2}. \tag{33.29}\]
This is the dictionary from the global Sweet–Parker layer to the local tearing variables.

Note

The plasmoid instability is not a different phenomenon from tearing. It is the tearing instability of the Sweet–Parker current sheet itself.

Constant-\(\psi \) and nonconstant-\(\psi \) limits. For a Harris-like reversing sheet the outer ideal solution gives (33.12)

\[\Delta ' a = 2\left (\frac {1}{ka} - ka\right ), \tag{33.30}\]
so for wavelengths longer than the current sheet thickness \(ka<1\) are tearing-unstable while the short-wave branch \(ka>1\) is stable.

The constant-\(\psi \) FKR scaling may be written schematically as

\[\gamma \tau _{A,a} \sim S_a^{-3/5}(\Delta ' a)^{4/5}(ka)^{2/5}. \tag{33.31}\]
and so
\[\gamma \tau _{A,a} \sim S_a^{-3/5}(ka)^{-2/5}, \qquad ka\,S_a^{1/4}\gg 1. \tag{33.32}\]
For even smaller \(k\) (longer wavelength), there is another solution, which is the so-called non constant-\(\psi \) branch Ara et al. (1978).
\[\gamma \tau _{A,a} \sim S_a^{-1/3}(ka)^{2/3}, \qquad ka\,S_a^{1/4}\ll 1. \tag{33.33}\]
The fastest mode sits at the crossover,
\[ka\,S_a^{1/4}\sim 1,\]
which implies
\[k_{\max }a \sim S_a^{-1/4}, \qquad \gamma _{\max }\tau _{A,a}\sim S_a^{-1/2}. \tag{33.35}\]
Returning to global variables using (33.28),
\[k_{\max }L \sim S^{3/8}, \qquad \gamma _{\max }\tau _{A,L}\sim S^{1/4}, \qquad \tau _{A,L}\equiv \frac {L}{V_A}. \tag{33.36}\]
So the instability is fast on the global Alfvén time even though it remains slow on the local sheet time. That is why high-\(S\) resistive MHD does not end in one monolithic Sweet–Parker layer.

Self-regulation by plasmoids. Once plasmoids appear, one should no longer think in terms of one sheet of length \(L\). Each interplasmoid segment of length \(\ell \) has its own local Lundquist number

\[S_\ell \equiv \frac {\ell V_A}{\eta _m}.\]
If \(S_\ell \) is still too large, that segment tears again and breaks into shorter sheets. The nonlinear state therefore resembles a hierarchy of near-marginal current layers joined by islands or flux ropes. This is one of the most important modern lessons of reconnection theory: within resistive MHD itself, a sufficiently large Sweet–Parker sheet is driven away from the laminar \(S^{-1/2}\) bottleneck.

33.5 Experimental perspective

Why the experiments matter. Reconnection is an unusually good example of a problem where laboratory experiments are not just illustrative. They are diagnostic of the theory itself. The theory predicts a coupling of global geometry, local current-sheet thickness, downstream pressure, and eventually kinetic physics. A controlled experiment can measure all four.

MRX and the generalized Sweet–Parker picture. On the collisional side, the Magnetic Reconnection Experiment (MRX) provided the classic test of the Sweet–Parker framework. Ji and collaborators showed that measured rates are broadly consistent with a generalized Sweet–Parker model once compressibility, downstream pressure, and an effective resistivity are included (Ji et al., 1998, 1999). This connects directly to (33.6)–(33.9): the pressure built in the center of the sheet is not a side issue. It can throttle the exhaust and thereby reduce the inflow rate.

From collisional layers to electron kinetics. In more weakly collisional regimes the simple scalar-pressure closure fails. Experiments on VTF and related theory showed that trapped-electron dynamics, electrostatic potentials, and pressure anisotropy can enter the electron force balance directly (Egedal et al., 2004, 2005, 2008, 2013). This is exactly the bridge from the present lecture to the generalized Ohm’s law and kinetic-MHD discussions elsewhere in the notes: the statement “flux freezing is broken in a tiny region” survives, but the identity of the non-ideal term changes.

TREX, asymmetric drive, and pileup. On TREX and the Big Red Ball, strongly driven asymmetric reconnection showed that shock formation and magnetic pileup can regulate the normalized rate (Olson et al., 2021; Greess et al., 2022). That result is conceptually important because it says the same thing as the flux-tube thought experiment at the start of the lecture: the outer drive matters. One cannot understand the local layer without understanding what the external magnetic configuration is trying to do. TREX also reported evidence for collisionless plasmoids and electron-scale current layers controlled by kinetic orbits rather than classical resistive broadening (Olson et al., 2016; Greess et al., 2021).

Solar-geometry experiments. The Big Red Ball program also produced a particularly attractive bridge back to solar and heliospheric plasmas. Peterson et al. observed laminar and turbulent plasmoid ejection in a laboratory Parker-spiral current sheet (Peterson et al., 2021). That is a laboratory realization of the same kind of long, driven current-sheet geometry that later appears in the heliosphere and in streamer-like structures. It is therefore a natural companion to the Y-point and helmet-streamer discussion above. Likewise, the line-tied screw-pinch experiments (Brookhart et al., 2015, 2017) make contact with coronal reconnection in geometries where footpoint motion, loss of equilibrium, and current-sheet formation are inseparable.

33.6 What to remember

Takeaways
1.
Two like-current flux tubes attract. The thin current sheet between them is a reactive layer, not the original source of the drive.
2.
The pressure at the X line is elevated to \[ p_X \simeq p_{\rm up} + \frac {B_0^2}{2\mu _0}, \] so the sheet converts incoming magnetic pressure into plasma pressure.
3.
The pressure drop away from the X line helps launch an Alfvénic exhaust, while the uniform reconnection electric field gives the matching condition \[ E_z \sim v_{\rm in}B_0 \sim \eta _\Omega J_z. \]
4.
Sweet–Parker reconnection is slow because a long thin layer forces \[ \delta /L\sim S^{-1/2}, \qquad v_{\rm in}/V_A\sim S^{-1/2}. \]
5.
Reconnection is non-local. The inner layer changes topology, but the outer ideal field determines the global drive, the sheet length, and in linear theory the sign of \(\Delta '\).
6.
At large Lundquist number the Sweet–Parker sheet is itself tearing-unstable and breaks into plasmoids.

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Problems

Problem 33.1.
Starting from (33.5), repeat the X-line pressure estimate when a uniform guide field \(B_g\) is present upstream and downstream. Show explicitly why \(B_g\) cancels from the leading-order pressure jump.
Problem 33.2.
Use (33.9) to derive the exhaust speed when the downstream pressure satisfies \(p_{\rm out}=p_{\rm up}+\alpha B_0^2/(2\mu _0)\). How does the result change as \(\alpha \rightarrow 1\)?
Problem 33.3.
Starting from the Harris-sheet expression (33.12), show that \(\Delta '>0\) for \(ka<1\) and explain physically why this corresponds to an attractive outer-region interaction.
Problem 33.4.
Combine (33.18) and (33.19) for a compressible exhaust with density ratio \(\rho _{\rm out}/\rho _{\rm in}=C\). How do the Sweet–Parker thickness and inflow rate change with \(C\)?
Problem 33.5.
Starting from (33.32) and (33.33), recover the scalings (33.36) for the fastest plasmoid mode.