The knot the field cannot untie

Flux freezing is usually stated as a conservation law: the flux through any loop moving with a perfectly conducting fluid does not change. Stated that way it sounds like a convenience.

Its real content is a prohibition, and the prohibition is much stronger than the conservation. If no loop’s flux can change, then no motion of the fluid can alter which parcel of it is magnetically connected to which. The topology of the field is fixed, for ever, whatever anybody does to the fluid.

Where the energy goes instead #

Push two of those flux systems toward each other. They cannot merge, so what happens is that the null is squashed.

The current density is not something added to the picture. The field of a squashed null has a curl where the symmetric one does not, and Ampère’s law then reads a current off it.

The circulation of B round a closed path counts the current threaded through it, and that relation is what turns a statement about the field’s shape into a statement about where the current is. A field that has been twisted has currents in it, those currents dissipate, and the energy stored in the twist is what reconnection releases — so the geometry and the energy are the same fact counted two ways.

So a frozen-in field responds to being compressed by accumulating energy in a sheet, and by definition it has no way of releasing it. That is the situation the Sun’s corona is in more or less permanently: convection at the surface shuffles the footpoints of coronal loops about, the loops are dragged into increasingly stressed configurations, and none of the stress can relax.

What freezing does when the fluid is squeezed as a whole is multiply the field by one over the square of the radius. A stellar core collapsing by a factor of a hundred thousand amplifies its field by ten billion — which is where a neutron star’s field comes from, and it is a conservation law rather than a dynamo.

It is worth saying where that energy is. It is in the field itself — a magnetic field carries an energy density B²/2μ₀ — so squashing the null is not storing energy in a spring somewhere else; the stored energy is spread through the volume the compressed field occupies.

The only way out #

Perfect conductivity is an idealisation, and the departure from it is a magnetic diffusivity $\eta$. Where the field varies over a length $\ell$, the field diffuses through the plasma in a time $\ell^2/\eta$.

For a coronal loop ten thousand kilometres across, that time is $10^{14}$ seconds — three million years. So diffusion cannot be the answer at the scale of the structure.

But the sheet is not the scale of the structure. If the current sheet is thin enough, diffusion through it is fast, and the field can change its connectivity there while remaining perfectly frozen everywhere else.

That the field diffuses at all is the ordinary irreversible spreading an equation that only runs forwards describes, with a magnetic diffusivity in place of a thermal one — which is why the time goes as the square of the length and why a thin layer is so much faster than a thick one.

The Lundquist number $S = L v_A / \eta$ compares the two, and it is enormous in every astrophysical plasma: $10^{13}$ in the corona, $10^{16}$ in the Earth’s magnetotail. That is why the ideal description works everywhere and why the exception is so small.

The rate, and why it is wrong #

Sweet and Parker did the balance in the late 1950s. Plasma flows into a sheet of length $L$ and thickness $\delta$, is accelerated out along it at the Alfvén speed, and carries the field with it; conservation of mass and of flux then fix $\delta$ and the inflow speed together.

The direction of the failure is the informative part. A better conductor reconnects more slowly, because a better conductor makes the sheet thinner and less plasma can be pushed through it. Every improvement in the plasma’s conductivity — every step further into the regime where the ideal description is a better approximation — makes the discrepancy worse.

That is not the shape of an error in a coefficient. It is the shape of a missing mechanism.

Why the balance gives a square root #

The Sweet–Parker result is worth deriving rather than quoting, because it is four lines and the square root is not obvious.

Plasma enters the sheet from above and below at a speed $v_{\text{in}}$ across a length $L$, and leaves along the sheet through two openings of thickness $\delta$. Mass conservation, for an incompressible fluid, gives $v_{\text{in}} L = v_{\text{out}} \delta$.

The outflow speed is fixed by the physics rather than chosen: the reconnected field lines are sharply bent and their tension flings the plasma along the sheet, and working the momentum balance through gives $v_{\text{out}} = v_A$, the Alfvén speed, whatever the resistivity is.

The field has to diffuse across the sheet as fast as it is carried in, or the sheet would thicken without limit. That gives $v_{\text{in}} = \eta/\delta$.

Three equations, three unknowns. Eliminating gives $\delta = L/\sqrt{S}$ and $v_{\text{in}} = v_A/\sqrt{S}$, and the time to process a structure of size $L$ is $L/v_{\text{in}} = \tau_A\sqrt{S}$ — the geometric mean of the Alfvén crossing time and the resistive diffusion time.

The square root is the signature of a geometric mean, and geometric means turn up wherever a fast process and a slow one are forced to run at the same rate through a boundary whose size is free to adjust. The sheet thins until the two agree, and the answer sits between them.

What has been proposed instead #

Three families of answer, and they share a strategy: stop the layer from being one long thin sheet.

The sheet breaks up. A Sweet–Parker layer with an aspect ratio above about $10^4$ is unstable to tearing, and it fragments into a chain of magnetic islands separated by many short sheets. The reconnection rate then becomes almost independent of the Lundquist number, which is what the observations require. This is the plasmoid instability, and it is now the standard account for high-$S$ systems.

The layer is thinner than the plasma is a fluid. When $\delta$ falls below the ion inertial length — the scale at which the ions stop following the field and only the electrons do — extra terms in the generalised Ohm’s law take over, the geometry opens out, and the rate rises to a few per cent of the Alfvén speed regardless of $S$. This is what spacecraft measure directly in the Earth’s magnetosphere.

The inflow is turbulent. A turbulent flow field wrinkles the sheet, multiplying its area, and the reconnection rate follows the turbulence rather than the resistivity.

A magnet falling through a copper tube is the laboratory version of the same competition, at a scale where every number is known. The field wants to move, the conductor resists the change, and the balance between them is a magnetic Reynolds number — the identical quantity that decides whether reconnection is slow or fast, measured here in a demonstration anybody can do.

The measurement that settled which mechanism #

For fifty years reconnection was inferred rather than watched. What was observed was the consequence — a flare, a substorm, a sawtooth crash — and the layer itself, being of negligible volume and usually a hundred million kilometres away, was never in the data.

That changed in 2015, when four spacecraft were flown in a tetrahedron a few kilometres on a side through the Earth’s magnetopause, sampling the electric and magnetic fields and the particle distributions fast enough to resolve the electron scale. The mission’s whole design is a statement about the problem: the layer is small, so the instrument has to be smaller, and it has to be in several places at once because a single spacecraft cannot tell a structure moving past it from a structure changing in time.

What it found was that the ions do stop following the field where the theory says they should, that the electrons continue to for a further distance, and that the region where the frozen-in condition fails for the electrons is a few kilometres across in a magnetosphere sixty thousand kilometres deep. The rate measured there is a few per cent of the Alfvén speed — a hundred times faster than Sweet and Parker allow and consistent with the collisionless account.

The corona remains inferred. Nothing has been flown through a solar current sheet and nothing will be; what is available there is the timing of the release, the geometry of the loops before and after, and the spectrum of the accelerated particles. That the terrestrial case and the solar case are the same mechanism is an argument from similarity rather than a measurement, and it is the standing assumption of the subject.

What it is used to explain #

Field lines are a choice rather than a discovery, and this is the essay where that matters most. What is physically defined is the connectivity — which parcels of fluid are joined by a line — and reconnection is precisely a change in that. So the phenomenon is named after a feature of a representation, which is unfortunate, and the underlying statement is about the topology of the field rather than about any drawn curve.

A solar flare releases $10^{25}$ joules in a few hundred seconds, which is the magnetic energy of a stressed active region and cannot be anything else — there is no other reservoir of that size on that timescale. A coronal mass ejection is the same event with the reconnection occurring beneath an erupting flux rope. A geomagnetic substorm is reconnection in the Earth’s magnetotail, converting energy stored from the solar wind. And a tokamak’s sawtooth crash is reconnection inside the confining field, redistributing the core plasma on a timescale far shorter than resistive diffusion allows.

An accelerated charge radiates, and reconnection is the most efficient particle accelerator in the solar system — so the events are seen by their radiation rather than directly. A solar flare’s hard X-rays are the signature, and the spectrum reports the energy the particles reached rather than anything about the geometry that produced them.

What it costs #

The field has been treated as a single object with no frame attached. Whether a given region carries an electric field as well depends on who is looking, and two combinations of E and B are the same for every observer — which is what makes a null point a frame-independent object and a current sheet a real one.

Two dimensions throughout. Every figure here is a slice, and reconnection in three dimensions has no neutral points in general, no separatrices, and a much richer classification of where connectivity can change. The two-dimensional picture is a special case and is the one all the intuition comes from.

The plasma has been a single conducting fluid. Ions and electrons have different masses, different gyroradii and different collision rates, and inside a thin layer those differences are the physics rather than a correction.

The resistivity has been a number. In a hot collisionless plasma it is not: the effective resistivity in a reconnection layer is set by wave–particle interactions, and the quantity called “anomalous resistivity” is a placeholder for a mechanism rather than a measured property of the material.

A changing flux through a circuit drives an emf, and ideal magnetohydrodynamics is that law with the resistance set to zero. Everything in this essay is what happens when a small resistance is put back: the flux is conserved almost everywhere and not quite, and the “almost” is confined to sheets thin enough that the resistance matters there and nowhere else.

What actually triggers it #

The essay has treated the current sheet as something the fluid motion builds. In the largest events it is better read the other way round: the field loses equilibrium first, and the sheet forms because it has to.

A coronal flux rope is held down by the field arching over it. As the footpoints are sheared the rope gains twist, and two ideal instabilities are waiting.

The first is the kink. A twisted rope is stable while the twist is modest and buckles into a helix once the total twist exceeds something between two and a half and three and a half turns — the precise threshold depending on how the twist is distributed. The buckling is ideal, meaning it changes no connectivity and needs no resistivity, and it can happen in an Alfvén time.

The second is the torus instability, and it is the one that decides whether an eruption escapes. A current ring wants to expand: its own field pushes outward on itself, a force that falls off with the ring’s radius more slowly than the confining field does if the confining field falls off fast enough. The criterion is written as a decay index — how steeply the overlying field weakens with height — and above about 1.5 the rope cannot find a new equilibrium at any height and runs away.

What follows is forced. The rope rises, the field beneath it is stretched into two opposed legs, and a current sheet forms between them whether anybody wants one or not. Reconnection in that sheet then cuts the tethers still holding the rope down, which lets it rise faster, which stretches the sheet further. The eruption and the reconnection drive each other, and asking which caused which is asking for a first cause in a loop.

This is the standard flare model and it explains the observed geometry rather well: two ribbons of brightening at the footpoints of the reconnected field, separating as the reconnection works its way up through the sheet into progressively higher and less-sheared field, with an arcade of cooling loops filling in behind them. Every one of those features is a prediction about where rather than about when, and the model is good at where.

What remains genuinely open is the last question anybody wants answered. The instabilities have thresholds, the thresholds depend on a coronal field nobody can measure directly — magnetic field in the corona is inferred from the photosphere below and extrapolated — and the extrapolation is exactly the quantity whose error decides whether the criterion has been met. That is why solar flare forecasting is statistical.

The particles, and why they are the awkward part #

A flare’s most conspicuous output is not heat. A large fraction of the released energy — estimates run from ten to fifty per cent — leaves as electrons and ions accelerated far above thermal energies, and those particles carry a power-law spectrum rather than any temperature.

That is already a strong statement about the mechanism, because a power law is what a scale-free acceleration process produces and a thermal distribution is what collisions produce. Whatever is accelerating these particles is not equilibrating them.

The awkwardness is arithmetic. Inferring the electron flux from the hard X-rays a flare emits when those electrons strike the dense chromosphere gives something like $10^{36}$ electrons a second above twenty keV, sustained for tens of seconds. The coronal loop supplying them contains perhaps $10^{37}$ electrons in total. So the acceleration region must process the entire electron population of the loop several times over during one flare, which means the electrons are not merely accelerated but resupplied — drawn up from the chromosphere, accelerated, sent back down, and replaced.

That is the number problem, it has been known since the 1970s, and it is why a picture in which a small diffusion region accelerates particles directly cannot be the whole story. The candidates that survive it all involve acceleration spread over a volume much larger than the layer: many small sheets in a fragmented plasmoid chain, repeated crossings of a shock at the top of the arcade, or turbulence filling the loop and accelerating stochastically.

Each of those makes a different prediction about how the spectral index relates to the flare’s size and how the acceleration site’s location moves during the event, and imaging spectroscopy in hard X-rays is the instrument that tells them apart. The answer is not yet settled, and it is settled least where the energy is largest.

Where the model stops #

Nothing here computes when a flare happens. The stress accumulates continuously and releases suddenly, and what triggers the transition is an instability whose threshold depends on the whole configuration. Every account of reconnection describes the release and none predicts the moment.

The energy released is not all of the stored energy. A field cannot relax to zero: it must keep whatever flux threads the boundary, and the minimum-energy state consistent with that is a force-free field rather than no field. What is available is the excess over that minimum, and estimating it requires knowing the boundary.

And the scale separation is extreme enough to be a computational problem in itself. A simulation resolving a three-metre layer inside a ten-thousand-kilometre structure needs $10^7$ cells across one dimension of the sheet, which is why almost every published simulation runs at a Lundquist number many orders of magnitude below the real one and extrapolates.

A straight current’s field circles it and falls as $1/r$, and a current sheet is a plane of such wires side by side. That is the object reconnection happens at: two regions of oppositely directed field meeting, with a sheet of current between them, and the whole question is how thin that sheet gets before the resistance can act.

What the pictures cannot show #

The X-point figures draw a field and a current density, and the drawing has no plasma in it. What is physically happening is that a fluid is being pushed, is carrying a field, and is piling up — and the fluid, its density, its pressure and its flow are all absent from a figure whose subject is the geometry of the field alone.

Nor can any of them show the layer. Drawn to scale inside the coronal structure it belongs to, the diffusion region would be three metres in ten million: less than a millionth of a pixel. Every published picture of reconnection, including these, exaggerates the layer by five or six orders of magnitude, and the exaggeration is not a stylistic choice — a figure at true scale would show two flux systems approaching and nothing else at all.

Where this ladder goes next #

Two rungs stand on flux-freezing. The first established the conservation law and its consequence for a collapsing star. This one finds the prohibition hiding inside the same law, and finds that the interesting physics is where the law locally fails.

The habit worth carrying away concerns idealisations that are almost exactly right. When an approximation holds to one part in $10^{13}$ everywhere except in a region of negligible volume, expect that region to decide the behaviour. It is the same shape as the boundary layer in a viscous flow, the shock in a compressible one, and the contact patch of a rolling wheel: a description excellent over the whole domain, and a small region where it fails carrying every rate the problem has.

What is left on this ladder is the state a stressed field relaxes toward. A field that has released everything it can, subject to keeping the flux through its boundary and the helicity it started with, is a force-free field with a particular constant — and the fact that real relaxing plasmas reach it, in laboratories and in the corona alike, is a result about which invariants survive a violent process and which do not.