The current no particle carries

A plasma confined by a magnetic field is pushed outwards by its own pressure and held in by the field, and the balance between the two is written in one line of fluid mechanics:

$$\mathbf{j}\times\mathbf{B} = \nabla p.$$

The pressure gradient points inwards, towards the hot dense centre, and the only way a magnetic field can push back against it is through a current flowing across the field and across the gradient. Crossing both sides with $\mathbf{B}$ gives the current that the balance needs: $\mathbf{j}_\perp = \mathbf{B}\times\nabla p/B^2$. Its size is fixed by the pressure gradient and the field strength, and nothing else.

The drift that does not care what the charge is built up the motion of a charged particle in a magnetic field from the other end: a fast circle, and a slow drift of its centre whenever something other than a uniform magnetic field acts. So the natural question is which drift carries this current. In a uniform field with no electric field and no gravity, the answer is none of them. Every guiding centre sits still. The pressure gradient is simply more particles on one side than the other, and no force acts on any particle to move its centre anywhere.

The current is there regardless. Its magnetic field can be measured, and in a fusion device the measurement is routine. The resolution is that a current is not the same thing as a flow of guiding centres, and the difference is where a plasma’s magnetism comes from.

Circles that go nowhere #

Picture the plasma edge from along the field. Each ion goes round a circle, all of them in the same sense — clockwise, for positive ions in a field pointing out of the page — and the circles are more numerous towards the dense side.

Stand on the dashed line and watch the ions pass. A circle centred to the left of the line meets it on its right-hand side, where the motion is downwards, and it does so at both of its crossings. A circle centred to the right meets it on its left-hand side, moving up. If there were as many circles on each side the two would cancel. There are not, so more ions cross the line going down than going up, and a net downward flow of positive charge is a current.

The sample drawn is small enough that the count could have come out either way, eleven down against seven up here, and that is why the average is the statement worth checking. For a Maxwellian spread of speeds and a density falling as $e^{-x/L}$, weighting every gyrophase and every speed by the density of guiding centres that would put an ion at the line gives a mean velocity of exactly $v_t^2/\Omega L$, where $v_t$ is the thermal speed and $\Omega$ the gyrofrequency. The figure computes that average numerically and compares it with the formula. The result is exact because weighting a Gaussian by an exponential shifts its centre without changing its shape.

That velocity is the fluid velocity of the ions, and it is called the diamagnetic drift, although nothing is drifting. Writing $n T$ for the pressure of one species, it is

$$\mathbf{u} = \frac{\mathbf{B}\times\nabla p}{q\,n\,B^2},$$

which is opposite for ions and electrons because of the charge in the denominator. The ions flow one way and the electrons the other, and since their charges are also opposite their currents add. The size is not negligible: at the edge of a fusion plasma at a hundred electronvolts, with the pressure falling over two centimetres in a three-tesla field, the ion fluid moves along the edge at nearly two kilometres a second, while every ion’s guiding centre stays where it is.

Why both species push the same way #

It is worth being clear about why the electrons help rather than cancel, because the argument is the whole reason a plasma is diamagnetic rather than merely magnetic.

A magnetic field does no work on a moving charge; it only turns it, and it turns positive and negative charges in opposite senses. Seen along a field pointing out of the page, an ion goes round clockwise and an electron anticlockwise. A clockwise positive charge and an anticlockwise negative charge are the same current, circulating clockwise, and a clockwise current seen from above produces a field pointing into the page, against the applied one. Every gyrating particle, of either sign, is therefore a small magnet opposing the field it is in. The sense of rotation is fixed by the sign of the charge, and the sign of the charge cancels out of the current it makes.

That is a statement about any charge in any field, and it is the classical root of diamagnetism in general: the orbiting response of a charge to a field is always to oppose it, which is the same instinct Lenz’s law describes for induced currents in a conductor. In a plasma it is not a small correction to anything. Each particle’s moment is its perpendicular kinetic energy divided by the field, so the magnetisation is set by the plasma’s thermal energy, which in a hot plasma is large.

The two species contribute in proportion to their pressures. In a plasma where ions and electrons share a temperature, they carry equal shares of the current, even though an electron’s circle is forty-three times smaller than a proton’s at the same temperature. The electron’s smaller circle is traversed faster, and the moment depends only on the energy. The ion fluid flows along the edge one way, the electron fluid the other, and an observer who measured only the electrons’ motion would find half the current and a flow opposite to the ions’, with no particle of either kind actually travelling along the edge.

The same current, counted twice #

The claim that circles can make the fluid’s current is checkable in detail, not only at one point.

The dots add up, at each position across the edge, the velocities of every ion whose circle passes through that position, with the guiding centres distributed like the density and the speeds Maxwellian. The line is the fluid theory’s $p'/B$. They agree to within six per cent of the peak everywhere, and the small difference at the steepest point, 0.963 against 1.000, is the curvature of the density profile across a gyroradius, which the fluid theory neglects and the particle count does not. Make the edge wider compared with the gyroradius and the difference shrinks as the square of the ratio.

What makes this work is a fact about each circle on its own. A charge going round a circle is a current loop, and a current loop far away is a magnetic moment. Each gyrating particle carries a moment $\mu = mv_\perp^2/2B$, and because a positive charge goes round one way and a negative charge the other, the moment points against the field for both. A plasma is a collection of small magnets all aligned to oppose the field they sit in, with a magnetisation per unit volume

$$\mathbf{M} = -\frac{p_\perp}{B}\,\hat{\mathbf{b}},$$

where $p_\perp$ is the pressure from motion across the field. A uniform magnetisation produces no current inside a body, since neighbouring loops cancel wherever they touch. A magnetisation that varies does not cancel, and the uncancelled part is the bound current $\nabla\times\mathbf{M}$, the same current that flows round the surface of a bar magnet. For a pressure that varies across a uniform field, $\nabla\times\mathbf{M}$ is exactly $\mathbf{B}\times\nabla p/B^2$. The current a plasma needs to hold up its pressure is its own magnetisation current.

The hole the plasma digs #

A magnetisation that opposes the field weakens it. That is diamagnetism, and in a plasma it is large enough to matter.

The balance $\mathbf{j}\times\mathbf{B} = \nabla p$, with Ampère’s law $\nabla\times\mathbf{B} = \mu_0\mathbf{j}$ and a field that is straight, integrates to a statement about pressures: $B^2/2\mu_0 + p$ is the same everywhere across the field. The plasma’s pressure and the field’s own magnetic pressure add to a constant, so where there is plasma there is less field.

The field is found here the long way, by starting in the vacuum outside the slab and integrating Ampère’s law inwards with $j = p'/B$ at each step, and the result is checked against pressure balance to a part in a million. The ratio of plasma pressure to magnetic pressure, $\beta$, is the natural measure. Quoted against the field outside, a $\beta$ of 0.2 lowers the central field by eleven per cent. At 0.95 it lowers it to 0.22 of the outside value, and the local $\beta$ at the centre, measured against the field actually there, is 19: the plasma has pushed almost all of the field out of the region it occupies.

In that limit the plasma behaves like a superconductor expelling a field, and the analogy is close in effect and distant in mechanism. A superconductor expels the field because of a quantum condensate whose currents are set by the field itself. A plasma expels it because its particles gyrate, each one a small opposing magnet, and the more energy they carry across the field the stronger the opposition.

The drifts run backwards #

The field inside the slab is no longer uniform, and a non-uniform field does move guiding centres. A gyrating particle in a field that is stronger on one side has a smaller circle there, so its orbit does not close; its centre creeps along the direction perpendicular to both the field and its gradient. That drift down the gradient of the field’s strength carries charge, and it is fair to ask whether in a high-pressure plasma it takes over the job.

It does the opposite. The drift velocity of each particle is proportional to its perpendicular energy and to the field gradient, and summed over the particles it gives a current $p\,B'/B^2$. The magnetisation current, now with both $p$ and $B$ varying, is the derivative of $p/B$. Adding the two gives $p'/B$, the fluid current, exactly. But pressure balance makes $B'$ opposite in sign to $p'$, so the drift current flows against the total.

The share of the total carried by the drifts is $-\mu_0 p/B^2$, which is minus half the local $\beta$, and the figure checks that at every point across the slab. In a tokamak, with $\beta$ of a few per cent, the drifts carry a per cent or two of the current, in the wrong direction. At the steepest point of a slab with a vacuum-field $\beta$ of 0.8, where the local $\beta$ is nearly one, they run backwards at 47 per cent, and the magnetisation current carries half as much again as the total to make up for them.

This is the most useful fact in the subject for anyone moving between the particle and fluid pictures. The guiding-centre velocity and the fluid velocity are different quantities, and the difference is not a small correction. A fluid velocity includes the diamagnetic flow, which no guiding centre makes; a sum of guiding-centre drifts omits the magnetisation current, which can be the whole of the answer. Adding the drifts and forgetting the magnetisation gives a current of the wrong sign.

Weighing a plasma with a loop of wire #

Diamagnetism turns into an instrument because the flux the plasma pushes out can be measured with a single loop of wire.

A loop around the plasma, perpendicular to the field, encloses a magnetic flux. When the plasma is heated, its magnetisation pushes some of that flux out, and the change induces a voltage in the loop whose time integral is the change in flux. At low pressure, integrating pressure balance across the cross-section gives the change as $-\mu_0 W_\perp/B_0$, where $W_\perp$ is the thermal energy per unit length in motion across the field and $B_0$ is the applied field.

A column half a metre in radius at a peak pressure of one atmosphere in a three-tesla field holds 79 kilojoules per metre of perpendicular thermal energy and pushes out 33 milliwebers of flux, out of nearly sixty webers within the drawing. That is a part in two thousand, and a loop and an integrator measure it easily. The approximation and the exact result are indistinguishable at this pressure, and even at four atmospheres, with a central $\beta$ of 0.11, they differ by less than two per cent.

Fusion experiments use exactly this, the diamagnetic loop, as one of their standard measurements of how much energy the plasma holds, alongside a very different one that infers the energy from the plasma’s equilibrium shape. In a tokamak the measurement needs a correction, because the current driven along the plasma also produces a field component that adds to the applied field, a paramagnetic effect of the opposite sign. The two effects are separated by combining the loop with measurements of the field around the plasma’s cross-section.

The Earth does the same measurement on itself. During a geomagnetic storm, energetic ions and electrons injected into the inner magnetosphere form a ring around the planet, and their diamagnetism, together with their drifts in the dipole field, lowers the magnetic field at the Earth’s surface. A theorem due to Alexander Dessler, Eugene Parker and Kurt Sckopke makes the connection exact for a dipole field: the depression at the centre is proportional to the total energy of the trapped particles, whatever their distribution. Magnetometers at low latitudes record the depression, typically tens of nanotesla and several hundred in a large storm, and the index built from them is read as a measure of the ring’s energy.

Circles that are really helices, and a current no sample shows #

The orbit figure draws each ion as a circle, which is its motion projected along the field. In the plasma every ion also moves along the field at its thermal speed, so the true path is a helix, and the circles drawn are the shadows of helices sliding past each other out of the page. The projection loses nothing that matters for the current across the field and everything about the motion.

More importantly, the figure draws sixty ions, and sixty ions do not carry the current. The count at the dashed line came out eleven against seven, and another draw of the same density could have come out the other way. The diamagnetic current is a statement about an average over enormous numbers of particles, where the fluctuation in any finite count becomes negligible, and a picture can show only a finite count. The line and the dots of the second figure show the average; the circles show the mechanism; neither shows both.

Where the circles stop being enough #

The picture needs the gyroradius to be small. Magnetisation is a concept for particles whose circles are much smaller than the distance over which the pressure changes. When the edge is as steep as a gyroradius, the particle count and the fluid formula part company, as the figure’s six-per-cent difference begins to show, and at the very edge of a real plasma the ion orbits are as wide as the gradient and a kinetic calculation is needed.

A plasma in a box in full equilibrium has no magnetisation. A theorem of classical statistical mechanics, due to Niels Bohr and Hendrika van Leeuwen, says that a classical system in thermal equilibrium has zero magnetisation, which appears to contradict everything above. It does not, but the reason is worth having. In true equilibrium in a uniform field, the density is uniform, so there is no pressure gradient and no bulk current. At a wall, particles whose circles hit it skip along it instead of completing their orbits, and the wall a plasma builds against itself is where their current, flowing opposite to the magnetisation current, cancels it exactly. A magnetically confined plasma escapes the theorem because it is not in full equilibrium: it is held away from any wall with a pressure gradient that has to be continually maintained by heating.

Pressure need not be the same in every direction. The formulas used the perpendicular pressure. In a strongly magnetised plasma the pressure along the field can be different, and then the balance and the magnetisation both involve the two separately, with extra terms from the curvature of field lines.

Collisions have been left out. Nothing in the argument needed them, which is its strength: the diamagnetic current is a property of the distribution of particles at one instant, not of how they scatter. But collisions are what let particles cross the field, and a pressure gradient in a collisional plasma leaks outward at a rate set by the scattering. The current holds the pressure up; the collisions, and far more strongly the turbulence, are what let it slowly fall.

And the fields here are straight. In a torus the field lines curve, and the curvature drift adds to the gradient drift; neither changes the conclusion that the fluid current is the drifts plus the magnetisation current, but the bookkeeping is heavier and it is why tokamak equilibrium calculations are done with the fluid equations rather than by adding up particles.

Still open: how the current survives at the edge of a fusion plasma #

The field was followed from lines that never end, through the force that does no work and the loop that behaves like a needle, to the drifts that move a circle’s centre. Here the circles themselves, standing still, turn out to carry the current that holds a plasma up. The drifts are not wrong, but they are only half of the bookkeeping, and at high pressure the smaller and backwards half.

The habit worth carrying away concerns what a velocity describes. When a fast motion is averaged away, ask whether the average of the position moves at the same rate as the average of the velocity; if the fast motion is not uniform in space, it does not, and the difference is a flux. The diamagnetic flow is that difference for gyration, and the same distinction between the motion of centres and the flux of particles returns wherever a fast oscillation sits in a gradient — including the slow drift that a sound wave leaves behind in a fluid.

What is not settled is how the pressure gradient at the edge of a high-performance tokamak holds itself. In the mode of operation that fusion reactors are designed around, the pressure drops across a pedestal only a few centimetres wide, steep enough that ion orbits and the gradient are comparable in size, and the diamagnetic flows there shear strongly. That shear is believed to suppress the turbulence that would otherwise flatten the gradient, so the steep edge helps sustain itself, but the mechanism that sets the pedestal’s width, and so how much pressure a reactor can hold, is still worked out with large simulations and compared with measurement one machine at a time.