The twist a bottle of plasma has to have

The field that cannot get out described a conductor so good that its magnetic field lines are carried along with it, and said that the reasoning runs both ways: if the field cannot leave the plasma, the plasma cannot leave the field. A charged particle in a magnetic field spirals around a line, and it has no way to step across to the next line without a collision. So a plasma threaded by a strong field ought to be confined across the field, and it is. That is all a magnetic bottle has to work with. The problem is the other direction, because the particle runs freely along the line, and a line that ends at a wall delivers everything to that wall in microseconds.

Lines have to close, therefore. The field must have no ends. And the simplest closed line is a circle, so the obvious bottle is a ring: a long solenoid bent round on itself until its ends meet, with a field that runs round and round inside a doughnut-shaped vessel. Every line closes on itself and none meets a wall.

This obvious bottle fails faster than almost anything else in plasma physics. Understanding why it fails, and how twist repairs it, gives the shape of the machine called a tokamak, including the odd and narrow constraint its builders live with.

The drift nobody can switch off #

A solenoid bent into a ring has a field that cannot be uniform. The coils are closer together on the inside of the ring than on the outside, so the field is stronger near the centre of the doughnut, falling off as $1/R$ with distance $R$ from the central axis, and the lines themselves are curved. Both matter, because a particle spiralling around a line in a gradient does not stay on it.

The drift that does not care what the charge is dealt with what a force does to a gyrating charge: its orbit tightens on one side and loosens on the other, so the centre moves sideways at right angles to both field and force. A particle following a curved line feels a centrifugal force. A particle in a gradient feels the weakening field as a force of its own. In a torus these both point outward along the major radius and combine into one drift, in the vertical direction, of size

$$v_d = \frac{m\left(v_\parallel^2 + \tfrac12 v_\perp^2\right)}{q B R}.$$

At thermal energy that comes to about $2T/eBR$. For a 10 keV ion at ITER’s major radius and 5.3 T field, the answer is about 600 m/s. That is slow compared with its own thermal speed of a million metres a second. The sign carries the problem: the charge $q$ is in the denominator, so ions drift one way and electrons the other.

Taken alone, that drift would lose a particle to the wall in a few milliseconds, as in the straight line of the figure. Fusion needs seconds, but the plasma never gets as long as milliseconds. Something much faster happens first.

Three microseconds #

The separation of charge cannot simply accumulate. Once ions have collected at the top of the bore and electrons at the bottom, there is a vertical electric field across the plasma, and a vertical electric field in a toroidal magnetic field produces a drift of its own. Unlike the first, that $E \times B$ drift does not care about charge, so ions and electrons move together. Its direction is outward, along the major radius, and it carries the entire plasma with it.

The acceleration works out to about $2c^2/R$, where $c$ is the plasma’s sound speed. A plasma that starts at rest and accelerates at that rate covers the minor radius $a$ in about $\sqrt{aR}/c$. For ITER’s dimensions at 14 keV, $c$ is about a million metres a second and the time is about three microseconds. Note how badly this scales. A hotter plasma leaves sooner, since the sound speed rises, and a larger machine gains only as the square root of its size. No amount of extra field helps, since $B$ has cancelled. The bottle is about a million times too leaky, and the leak is not a matter of engineering.

This is the same instability of a heavy fluid over a light one that the salt that sinks through a stable sea showed for a different reason, with the outward centrifugal force of the field line’s curvature playing gravity. The plasma, trying to expand to where the field is weaker, is a dense fluid sitting above nothing.

Where a field line comes back to #

The repair came in 1951 from two directions. Lyman Spitzer, at Princeton, thought of it in connection with his stellarator; Igor Tamm and Andrei Sakharov, in Moscow, arrived at their own design at almost the same time. The drift, they each saw, is always vertical, but a particle’s position in the bore need not be. If the field lines also wind the short way round the bore as they go the long way round the torus, then a particle following its line spends half its time above the midplane and half below. Then a vertical drift moves it outward from the centre on one side and inward on the other, and the two halves cancel.

This winding is measured by a number with a misleading name. The safety factor $q$ is how many times a field line goes round the torus for each time it goes round the bore. It is a large number for a mild twist and a small number for a tight one.

Lines with a twist do not close after one circuit, and that changes what a bottle is. Follow one line, and note where it crosses a fixed cut through the bore on each pass.

If $q$ is a rational number it visits a finite set of points and closes on itself. If $q$ is irrational it never closes; its crossings fill in a circle, and the one line traces out an entire surface shaped like a doughnut’s skin. Nested inside one another, these surfaces fill the machine. Electrons run along a line many times faster than any drift, so a line that covers its surface shorts out any difference of potential across the surface, which is precisely what removes the electric field behind the three-microsecond loss. The irrational surfaces do this cleanly. The last curve to go is about what the same number theory does in a very different system, and the map involved there is one that a tokamak’s field lines obey too: where a small disturbance resonates with a rational surface, the surface tears into a chain of islands, and a large enough disturbance dissolves the tidy nesting altogether.

The drift continues throughout. Twisting has not stopped it, only made it average out. As the figure at the start shows, the guiding centre of a 10 keV ion at $q = 3$ traces a circle, displaced from its flux surface by the drift speed divided by its rate of circulation. That offset is about 1.3 cm, and it was integrated, not assumed. For electrons the offset has the opposite sign, and for particles travelling slowly along the line, the offset is much larger, tracing a shape known as a banana; those orbits are where most collisional leakage in a tokamak comes from.

Making the twist with a current #

Spitzer’s solution twists the field with external coils, wound helically. That is a stellarator, and it took sixty years of computing to design one well. Tamm and Sakharov’s solution is simpler. Drive a large current through the plasma, the long way round. That current has a magnetic field of its own, which wraps around the plasma as the field that wraps a current showed for any wire, and added to the toroidal field gives a helix. The plasma is the secondary of a transformer whose primary is a coil through the middle of the doughnut.

Soviet physicists named it tokamak, from the Russian for a toroidal chamber with magnetic coils. It was not much noticed until 1968, when the T-3 machine reported temperatures of about 1 keV and a confinement time far above any Western result. A British team travelled to Moscow in 1969 with a laser scattering diagnostic to check the claim, and it was confirmed, and laboratories worldwide converted their machines. This is a current-carrying plasma, and a current is what makes the current that squeezes what carries it a relevant warning.

Now a tokamak’s twist depends on its current. For a round bore, the edge safety factor is

$$q_a = \frac{2\pi a^2 B}{\mu_0 I R},$$

which gets smaller as the current rises. Current helps in other ways, since it heats the plasma by its resistance, and larger currents confine better, in the empirical fits. So builders want as much as they can have.

ITER’s 15 MA in a round bore would give $q$ near 1.1, which is unworkable. The bore is therefore stretched vertically, and with an elongation of about 1.85 the same current has a longer path round the bore and so less twist. The cost of elongation is a new instability of the column’s vertical position, controlled by active coils at a millisecond’s notice. This shows why modern tokamaks have D-shaped cross-sections and are not doughnuts, and the shape is the direct result of this one line in the figure.

Why the column kinks #

Too little twist fails to short out the drift. It would be natural to expect that more would always be better. It is not, and the limit is a mechanical one that anyone can feel by overwinding a rubber band: past a certain twist, it stops storing twist and throws a loop out sideways. A column of plasma carrying a current behaves in precisely this way.

Martin Kruskal and Vitaly Shafranov independently found the condition in the mid-1950s. Look at a straight column of length $L$ and radius $a$, carrying a current in a longitudinal field. Its lines are helices. A bend in the column with the same pitch as a line can move sideways without having to bend the line, and so without doing any work against the field’s tension, while the current’s own field pushes the bend further out. It grows in the time an Alfvén wave takes to cross the column, which the wave that does not know what the gas is made of puts at a few microseconds for a fusion plasma. Stability requires that no such helix fits: the line must turn by less than one full revolution along the length, so the twist has to be limited.

For a torus, the column’s length is $2\pi R$, and the condition becomes $q > 1$ at the edge. So the twist that the drift needs is bounded from below, and the current it comes from is bounded from above. In practice, things get worse well before $q = 1$. Tokamak operators find that as $q_a$ approaches about 2, tearing modes on rational surfaces begin to grow, and the plasma suddenly loses its thermal energy in a millisecond or so and its current in tens of milliseconds, in a disruption. At ITER’s scale that releases hundreds of megajoules against the wall and induces currents with forces of thousands of tonnes in the vessel. Avoiding and mitigating these events is one of the largest unsolved engineering questions in the design of a working machine.

Nature has run the same test in a different setting. Loops of plasma in the solar corona are threaded by a field and rooted in the dense surface below, and they twist as the surface moves. Their ends are held fixed, which stabilises some bends, so their limit is higher than in a free column, and calculations place it near two and a half turns. Loops carrying that much twist can kink and erupt, a sequence observed in some solar flares; the knot the field cannot untie deals with how reconnection allows such a field to lose its tangles. The same constraint appears at scales of a metre and of ten thousand kilometres because Alfvén waves carry the information in both cases and the geometry alone sets the limit.

The orbits that never get round #

The field in a torus is stronger on the inside than on the outside, by a ratio of about $(R+a)/(R-a)$. That makes the outside of every flux surface a magnetic well. As a particle follows its line from the outer side towards the inner side, it enters a stronger field, and its magnetic moment is conserved, so its motion along the line slows; the drift that does not care what the charge is introduced this adiabatic invariant and the mirror it produces. A particle with too little speed along the line stops and turns back. It never gets round the short way at all.

For a surface at minor radius $r$, the fraction of particles trapped this way is about $\sqrt{r/R}$, which is a third or more in the outer half of a modern machine. These particles bounce back and forth between two points on the outer side, and their vertical drift then does not average out by the simple argument, since they never reach the inner half. It moves them inward on the way one direction and outward on the way back. Seen in the poloidal cut, their path is a pair of nearly parallel arcs closed at the ends, which is what physicists call a banana, and its width is roughly $q\rho/\sqrt{r/R}$, with $\rho$ the gyroradius. For a 10 keV ion in ITER that is about two centimetres, nearly twice the offset of the circulating orbit drawn above.

This has two consequences that a builder cares about. Collisions now move a particle by a banana’s width at a time, which is why collisional leakage in a torus is many times what it is in a straight field; the theory of it is called neoclassical, and it gives a lower limit on the loss, below which turbulence is what is actually observed. The second is an outright gift. The trapped particles’ orbits on neighbouring surfaces do not quite balance when the density and temperature vary with radius, and the small mismatch drives a net current along the field. Nobody supplies this current. Called the bootstrap current, predicted in 1970 and measured in the 1980s, it can carry most of the plasma’s current in a suitably designed machine, so that a tokamak can in part make its own twist from its own pressure gradient.

A straight cylinder and a single value of q #

The guiding-centre drawing uses one speed and one drift, a circular bore, and a single value of $q$. A real tokamak has $q$ varying across its radius, typically near 1 at the centre and above 3 at the edge, and that shear turns out to be highly useful since neighbouring surfaces with different twist cannot exchange plasma in simple interchange motions. The plain $E \times B$ argument for three microseconds is a rough estimate of a very fast thing, with its numerical coefficient only approximate.

The figure for the kink limit is a straight cylinder with a uniform current. Real current profiles are peaked, and at the centre, where $q$ falls below 1 in many machines, a kink of the central core does grow. It is limited there and reorganises the core repeatedly in a cycle called the sawtooth. Above all, nothing here reaches the question of why plasma leaks across the surfaces much faster than collisions would predict. Turbulence does that; it sets the confinement time that Lawson’s criterion treats as a given number, and these drawings have nothing to say about it.

Still open: whether a current is the right way to twist #

A tokamak’s twist comes from a current driven by a transformer, and a transformer can drive a current for only as long as the flux in its core is changing. A tokamak is by nature a pulsed machine unless the current can be sustained some other way, and all the disruption problems above come from having a large current in the plasma to begin with.

A stellarator has neither problem, since its twist comes from coils. It has no net current, no disruption of that kind and no pulse length limit. Its price is three-dimensional shape: untwisted drifts must be cancelled by careful geometry, which early stellarators failed to do, so they lost particles far too fast. Wendelstein 7-X in Greifswald, which began operating in 2015, has coils whose shapes were computed by optimisation, and it has shown that such confinement of fast particles can be achieved in practice. Whether a stellarator or a tokamak, or some combination, gives the more practical power plant is a live argument, and neither has yet burned deuterium and tritium at scale.

A ring of magnetic field has no ends, and in it a curvature drift separates the charges and throws the plasma at the wall in microseconds. Twisting the lines makes the same drift average to nothing, with an integrated orbit that closes to within about a centimetre. Twisting them by more than one turn along the column makes the column itself kink. A tokamak is a machine that runs between those two limits, and its shape and its current are both set by them.