The twist that outlives the turbulence

The ZETA machine at Harwell was a ring of hot hydrogen with a large current driven round it, built in 1957 to see whether the current’s own magnetic field could squeeze a plasma hot enough for fusion. It became famous for the wrong reason: the neutrons announced in January 1958 turned out not to come from fusion at all. What it recorded over the following decade was more interesting and took much longer to understand. After a violent, turbulent start, each discharge settled into a quiet period, and in that quiet period the magnetic field running the long way round the torus had reversed direction near the wall — pointing one way along the axis of the plasma and the other way at its edge. Nobody had put that reversal there. It appeared on its own, discharge after discharge, whatever the details of how the plasma had been started.

A plasma is the best conductor there is, and a perfect conductor cannot change which field line joins which piece of it. A turbulent discharge ought therefore to end in whatever tangle its turbulence happened to leave. The ZETA plasmas did the opposite: they forgot their history and arrived at one particular shape. John Bryan Taylor explained why in 1974, and the explanation is a statement about which conserved quantities are fragile and which are robust, arrived at by asking which one a little resistivity cannot destroy.

Too many constraints, or none #

A field left to itself sheds energy. If nothing constrained it, the minimum-energy state would be no field at all, and a relaxing plasma would simply lose its magnetism. That is not what happens, because two things are held: the flux threading the plasma is fixed by the conducting wall around it — the wall is a loop the flux cannot get out of, holding it by the opposition to change any closed circuit shows — and something about the field’s internal structure is fixed too.

What that something is decides everything. In ideal magnetohydrodynamics, with no resistivity at all, the answer is far too much. Every thin tube of flux keeps its own identity, its linkage with every other tube and the twist of its own lines. That is an infinite set of constraints, one per tube, and an energy minimum subject to all of them can be any of the uncountably many force-balanced tangles consistent with the starting topology. The prediction would be that the final state remembers the initial one in every detail, which is exactly what ZETA contradicted.

With a finite resistivity, however small, the answer seems to be none: field lines can break and rejoin, the tube-by-tube constraints are gone, and nothing forbids the field decaying all the way. That also contradicts the experiment, which kept a strong field in a definite shape for long periods.

The resolution is that the infinite family of invariants does not break all at once. Reconnection happens in thin sheets where the gradients are steep — a sheet three metres thick inside a structure ten thousand kilometres across, in the solar case — and inside those sheets individual tubes are cut and spliced freely. But one member of the family is the sum over all of them, and a sum over the whole volume is barely affected by what happens in a small part of it.

The quantity that is spared #

The member is the magnetic helicity,

$$K = \int \mathbf{A}\cdot\mathbf{B}\;\mathrm{d}V,$$

with $\mathbf{A}$ the vector potential whose curl is the field. Its topological meaning — that for two thin closed flux tubes it counts how many times one winds through the other, multiplied by the two fluxes — is a result of vortex dynamics and knot theory, and what matters here is a simpler property: in a perfectly conducting plasma it is conserved, and it is gauge-independent as long as no field line crosses the boundary. Both are true inside a closed conducting wall.

What resistivity does to it can be written down. With resistivity $\eta$, energy and helicity change at

$$\frac{\mathrm{d}W}{\mathrm{d}t} = -\eta\int J^2\,\mathrm{d}V, \qquad \frac{\mathrm{d}K}{\mathrm{d}t} = -2\eta\int \mathbf{J}\cdot\mathbf{B}\,\mathrm{d}V,$$

in units with $\mu_0 = 1$, and the second is controlled by the first. The Cauchy–Schwarz inequality bounds $|\int \mathbf{J}\cdot\mathbf{B}|$ by $\sqrt{\int J^2}\sqrt{\int B^2}$, so the rate of helicity loss is at most $2\sqrt{\eta}\,\sqrt{2W}\,\sqrt{|\mathrm{d}W/\mathrm{d}t|}$. Suppose turbulence arranges to dissipate energy at some fixed rate, however small $\eta$ is — which is what turbulence does, by driving the current into ever thinner sheets until the resistivity can act. Then the helicity loss rate goes to zero as $\sqrt{\eta}$. In the limit of a very good conductor the energy can be lost at a finite rate while the helicity is kept.

The inequality can be seen working in the simplest possible model. Build a field out of many twisted modes of wavenumber $k = 1, 2, 3, \dots$ in units of the largest mode the container allows, each carrying some energy and as much helicity as its energy permits, which is the energy divided by the wavenumber. Let each decay resistively at its own rate, proportional to $k^2$.

With the energy per mode falling only as one over the wavenumber, most of the energy sits in the many short-wavelength modes, and those are the ones resistivity attacks first, at a rate a million times faster for mode 1,000 than for mode 1. The helicity of each mode carries an extra factor of one over its wavenumber, so almost all of it sits in the few longest modes, which resistivity barely touches. By the time half the energy has gone, 96.7 per cent of the helicity is still there. The same thing happens more strongly in real turbulence, where the short scales are continually replenished from the long ones by the cascade and drained at the bottom, so energy leaks away continually while helicity, which in magnetohydrodynamic turbulence tends to move to larger scales rather than smaller, is kept out of the sink.

This is called selective decay, and the word selective is the whole idea. Nothing about the dissipation prefers one quantity for its own sake. The two quantities simply live at different scales, and dissipation lives at the smallest.

The floor under the energy #

Keeping helicity forces a field to keep energy, because a field cannot be twisted without being a field. The same mode picture gives the bound directly. The energy of each mode is at least its wavenumber times the magnitude of its helicity, and the smallest wavenumber available is $k_1$, set by the size of the container. Adding up,

$$W \;\ge\; k_1\,|K|,$$

with equality only if every bit of the energy is in modes of the lowest wavenumber. For a given helicity there is a least energy the field can have, and there is exactly one way of having it.

The three spectra begin at very different places. The one with energy spread to short scales starts with 4.52 times the minimum energy for its helicity, the steepest with only 1.11 times. All three end at 1.000. They lose different amounts of energy on the way, at different times, and arrive at the same ratio because it is the floor: once the energy has dropped to the least that the remaining helicity allows, nothing more can go without helicity going too, and helicity is the thing that does not go.

So the prediction for a relaxed plasma is not “the field decays”. It is “the field decays until it reaches the minimum energy consistent with its helicity and its flux, and then stops”. That minimum is a variational problem, and it has an answer that can be written down.

The only shape at the bottom #

Minimising the energy $\tfrac{1}{2}\int B^2$ while holding $K$ fixed is a problem with one constraint and one Lagrange multiplier, of the same form as a system that minimises its energy less its temperature times its entropy because a reservoir fixes the trade between them. Varying $\mathbf{A}$ and setting the variation of $W - \tfrac{\lambda}{2}K$ to zero gives

$$\nabla\times\mathbf{B} = \lambda\,\mathbf{B},$$

with a single constant $\lambda$ for the whole volume.

The equation says the current, which is the curl of the field, runs exactly along the field everywhere. A current parallel to a field feels no force from it, so the relaxed state is force-free: the magnetic stress is perfectly balanced at every point, with the pressure of the field and its tension cancelling. The constant $\lambda$ is fixed by the ratio of helicity to flux. And the fact that it is one constant, not a function varying from field line to field line, is the fingerprint of having kept only one global invariant. Keeping every tube’s helicity would allow $\lambda$ to vary from tube to tube; keeping only the total forces it to be the same everywhere.

In a straight cylinder of radius $a$ the solution is a pair of Bessel functions, the same functions that give a drum its inharmonic overtones:

$$B_z = B_0\,J_0(\lambda r), \qquad B_\theta = B_0\,J_1(\lambda r).$$

That is the ZETA reversal. $J_0$ has its first zero at 2.405, so any relaxed state with $\lambda a$ larger than that has an axial field that crosses zero inside the wall and is reversed beyond. At $\lambda a = 3$ the crossing is at $r = 0.802a$. Nothing about the edge was specified; the reversal is simply the shape the equation gives once the helicity is large enough for the flux. At $\lambda a = 1.5$ the axial field falls from the axis to about half by the wall and never reverses, which is the regime ordinary tokamaks, with their strong imposed axial field, live in.

A field that turns as it goes out #

A more direct way to see the state is to follow the direction of the field rather than its components. On the axis the field points along the cylinder. Moving outward, the azimuthal component grows and the axial component shrinks, and the field direction rotates.

A turn past 90 degrees is the reversal. What the arrows add to the component plot is the sense of a continuous twist: the field lines on successive cylindrical surfaces are helices of steadily changing pitch, and between the surface where they run straight along the axis and the wall where they run partly backwards there is a surface where they go purely round the circumference.

That changing pitch has a name, magnetic shear, and it matters for more than description. A pinch with little shear is prone to kinking, where the whole column bends into a helix because a helical displacement can lower the field energy. The kink can grow only where the displacement fits the pitch of the field lines, and a field whose pitch changes rapidly with radius offers no single pitch for a displacement to fit across the whole column. That is part of why the relaxed state lasts. The instabilities that drove the relaxation have used up their free energy, and the state they left is sheared enough to resist starting over.

Two numbers that diagnose a pinch #

Experimentalists describe a toroidal pinch with two ratios that can be measured with coils outside it. The pinch parameter $\Theta$ is the azimuthal field at the wall divided by the mean axial field — how hard the current is being driven for the amount of axial flux inside. The reversal parameter $F$ is the axial field at the wall divided by its mean. A negative $F$ is a reversed edge.

For the Bessel state both follow from $\lambda a$ alone. The mean axial field over the cross-section is $2B_0 J_1(\lambda a)/\lambda a$, so $\Theta = \lambda a/2$ exactly, and $F = \lambda a\,J_0(\lambda a)/2J_1(\lambda a)$.

The theory’s prediction is that a relaxed pinch does not wander about this plane. Whatever it started as, it lies on the one curve. Set the current and the flux, which fixes $\Theta$, and $F$ follows: 0.64 at a pinch parameter of 0.8, zero at 1.202, and −0.63 at 1.4. That the reversed-field pinch appears only above $\Theta \approx 1.2$ was what ZETA had seen without a reason, and the value falls out of the first zero of a Bessel function.

The measured curves from reversed-field pinch experiments — the Madison Symmetric Torus in Wisconsin, RFX in Padua, and their predecessors — follow this shape closely and depart from it in a consistent direction: real plasmas are less strongly reversed than the Bessel model at a given $\Theta$. The departure is understood. Near a cold wall the plasma’s resistivity is high and the current density falls off, so $\lambda$ cannot really be constant right up to the wall; models that let it drop in a thin edge layer fit the data well. That is a correction to where the one-constant rule applies, not a failure of the rule in the interior.

The curve also ends. Taylor’s own analysis found that above $\lambda a = 3.11$, a pinch parameter of 1.56, the axisymmetric Bessel state is no longer the lowest-energy solution: a state with a helical distortion has less energy at the same helicity and flux. That boundary is drawn and not computed here. Experiments driven to high current have in fact found plasmas spontaneously organising round a single helical structure, which is at least in the spirit of the prediction, though the full story there involves more than the minimum-energy argument.

What the relaxation releases #

The argument also says how much energy relaxation frees, and this is where it connects with the question the knot the field cannot untie left open: a stressed field cannot release all of its energy, so how much can it release?

Take a simple starting state that is not relaxed — a uniform axial field with a uniform current density flowing along it, so the azimuthal field grows linearly from the axis. That state is described by its own pinch parameter $\Theta_0$. Compute its flux and its helicity, find the Bessel state with the same flux and the same helicity, and compare energies. The comparison needs one caution: in a straight cylinder with flux running along it, the helicity depends on the choice of vector potential at the wall, so both states are measured in one gauge with the axial potential zero on the wall, which is enough when the two share the same wall and flux.

A pinch started at $\Theta_0 = 2$ relaxes to $\lambda a = 2.25$ and releases 19.3 per cent of its magnetic energy. One started at $\Theta_0 = 4$ relaxes to $\lambda a = 2.76$, a reversed state, and releases 44.8 per cent. The released fraction climbs steadily with how hard the starting current was pushed, which is what should be expected: more current on the same flux is more twist, and more of the energy is in the short-scale structure the relaxation removes.

The threshold for reversal is exact in an unexpected way. The starting pinch relaxes to a reversed state for any $\Theta_0$ above 2.405 — the first zero of $J_0$ again, now as a starting pinch parameter rather than a value of $\lambda a$. The coincidence is an identity: the uniform-current pinch has a helicity-to-flux-squared ratio of $\Theta_0/2\pi a$, and at the zero of $J_0$ the Bessel state has a ratio of $\lambda/2\pi$, so the two meet exactly there. The figure finds the threshold numerically, by root-finding on the relaxed state, and checks that it lands on the zero.

The energy released goes into heat and flows. In a laboratory pinch the relaxation is not a single event but a repeated one. Resistive diffusion keeps pulling the current profile away from the relaxed shape, since the plasma is hotter and more conducting at the centre, and every so often a burst of instability snaps it back, releasing energy and redistributing current outward. The field reversal at the edge is maintained against resistive decay by exactly those bursts, which in reversed-field pinch research is called the dynamo, and which is a sequence of small Taylor relaxations.

Spheromaks, and why the corona twists one way #

The same principle builds a plasma with no coils threading it at all. A spheromak is a ball of magnetised plasma injected into a conducting can, carrying both a field that runs round the ball and a field that loops through it. It is formed by firing twisted plasma out of a coaxial gun, which pumps helicity in, and it relaxes into the lowest-energy state of that helicity for the can’s shape. Changing the gun’s details changes the formation transient, and the final state is the one the can and the injected helicity select. It is the purest demonstration of the idea: a magnetic confinement configuration whose shape is set by a variational principle rather than by any external coil.

The idea first appeared in astrophysics. Lodewijk Woltjer proved in 1958 that the minimum-energy state of a field at fixed helicity is a force-free field with constant $\lambda$, with the Crab Nebula’s field in mind. Taylor’s contribution was to see why a turbulent laboratory plasma would conserve the total helicity while losing the rest, which is what turned Woltjer’s theorem into a prediction of where a real plasma ends up.

In the solar corona the argument takes an important turn. Twist is continually being put into coronal fields by the rotation and shuffling of their footpoints in the photosphere, and relaxation cannot remove it. Energy can be dissipated in the corona — the proposal of Heyvaerts and Priest in 1984 was that braided coronal loops relax towards Taylor states and heat the gas as they do — but the helicity, by the argument above, accumulates. It has to leave somehow, and one way it can leave is by being carried out bodily. Coronal mass ejections take away twisted magnetic structures, and a line of argument in solar physics holds that removing accumulated helicity is part of what they are for: the Sun cannot dissipate its helicity, so it throws it away.

There is a signature of this in the sky. Twisted structures on the Sun show a hemispheric preference: filaments and active regions in the northern hemisphere tend to carry one sign of helicity and those in the southern the other. The rule is statistical, with many exceptions, but it persists from one solar cycle to the next, which is what a quantity the corona cannot destroy, fed in with a sign set by the Sun’s rotation, would be expected to do.

Where one constant stops being enough #

Only the total is conserved, and only approximately. The argument needs resistivity small enough that the helicity loss is negligible and turbulence strong enough to reach the minimum. A plasma too resistive loses helicity too; a plasma too quiet never finishes relaxing, and keeps more of its tube-by-tube structure than the theory allows. Both limits occur, and the theory says nothing about how long relaxation takes.

The wall matters. Helicity is gauge-invariant only inside a boundary that no field line crosses. The solar corona has no such boundary — its field lines are anchored in the photosphere — and the helicity there has to be replaced by a relative helicity measured against a reference field with the same footpoints. That version carries the same physics, but it can be injected and removed through the boundary, which is exactly what footpoint motions and ejections do.

Pressure has been ignored. A force-free state is a zero-pressure state. Real pinches carry plasma pressure, and in the corona the gas pressure is small but not zero. Adding pressure modifies the minimum, and in a toroidal device the geometry of the torus adds corrections the straight cylinder does not have.

And a force-free field with one constant is the minimum, not the typical state. Coronal field models that assume one $\lambda$ everywhere match observations less well than models in which $\lambda$ varies from one field line to the next — which is to say, the corona is not fully relaxed, and keeps more memory of how it was twisted than a laboratory pinch does.

A quiet end state standing in for a violent process #

Every figure here draws either the relaxed state or a model of decay that has no turbulence in it. The mode picture lets each wavelength decay on its own under resistivity, which is the simplest way to show that energy and helicity live at different scales, and it leaves out the thing that actually does the work in a pinch: the nonlinear transfer of energy from long wavelengths to short ones, through bursts of instability and reconnection lasting microseconds and filling the whole volume in three dimensions. Those bursts break the tube-by-tube constraints in sheets thinner than any drawing could resolve, and they are what a laboratory measures as sawteeth in the current and flashes of radiation. The figures show where the field ends up and why it cannot go further. How it gets there is a turbulent process that only simulations of the full equations, and the machines themselves, can show.

Still open: whether the corona ever finishes relaxing #

Frozen flux has now been followed through five arguments: the conservation law, the prohibition it hides, the stress that makes the field mechanical, the wave that stress carries, and here the state the field is left in when the prohibition is broken just slightly.

The habit worth carrying away concerns invariants that survive their own breaking. When a perfect conservation law is broken by a small effect, ask which combination of the broken invariants the small effect can reach, and expect the rest to survive. The perfect-conductor constraints fail tube by tube where reconnection happens, and the sum over all tubes is almost untouched because the failures are confined to thin sheets. The same shape of argument explains why a magnet falling down a copper pipe keeps its field while its kinetic energy goes into heat.

What is not settled is how far the corona gets. A laboratory pinch is stirred hard inside a closed conducting wall and relaxes in milliseconds; a coronal loop is stirred gently at its feet, is open to the photosphere, and may never be driven long enough in one direction to reach its floor. Whether coronal heating is Taylor relaxation, partial relaxation or something else, and how much of the helicity budget ejections actually carry off, are measured questions still being argued over, and the answer decides whether the one-constant state is where the solar field ends up or only the direction it is heading.