The field no symmetric flow can keep

A lump of copper a metre across, given a magnetic field and left alone, loses it in about two seconds. The currents that carry the field run into the copper’s resistance, their energy turns into heat, and the field goes with them. Scale the lump up and the loss slows down, as the square of the size, because a bigger conductor has more room for its currents and a longer way for the field to leak out — but it never stops. The Earth’s liquid iron core, 3,480 kilometres in radius, would lose its field the same way in roughly forty thousand years.

Rocks laid down three and a half thousand million years ago carry a record of the field they cooled in, and it was there then, at a strength not very different from today’s. So the Earth’s field is not a relic. Something is putting it back as fast as resistance takes it away, and the only thing in the core moving fast enough to do it is the iron itself, stirred by heat escaping from the deep interior and by the slow freezing of the inner core.

That a moving conductor can make a field is ordinary. A field is carried by a good conductor as if it were painted on, so a flow that stretches the conductor stretches the field lines and strengthens them, and a flow that folds them can pack more of them into a volume. Joseph Larmor proposed in 1919 that the Sun’s sunspot fields might be made this way. The trouble is that the obvious flows, the ones simple enough to calculate, refuse. Thomas Cowling found why in 1934, and the reason has nothing to do with how fast the iron moves.

A field that circles a ring #

Take a field that looks the same from every direction round some axis — the field of a bar magnet, or the dipole that dominates the Earth’s field, or anything else with that symmetry. Split it into two parts. The toroidal part points round the axis, like the lines of latitude on a globe. The meridional, or poloidal, part lies in the planes that contain the axis, and it is what a compass needle on the surface mostly feels.

Because of the symmetry, every meridional plane holds the same picture, and the meridional lines in one plane are the contours of a single function. That function measures how much flux passes through a circle around the axis at each point. It is zero on the axis, where the circle has shrunk to nothing, and it falls to zero again far away, where the field has died out. Between them it is positive.

A positive function that is zero on two boundaries has a maximum somewhere between. At the maximum the contours shrink round a point, and the meridional field — which runs along the contours with a strength set by how closely they are packed — is exactly zero there. In three dimensions the point is a ring encircling the axis, the O-ring, and every closed meridional line inside the conductor goes round it.

None of this depends on the field being a dipole or on the conductor being a sphere. The figure draws the slowest way a magnetised conducting sphere can decay, because it is a real solution with nothing arbitrary in it, but any symmetric field whose meridional lines close within the conductor has a ring like this one. It is the same kind of fact as the count of places where a handful of charges’ fields cancel: a statement about where a field must vanish, made before any of its details are known.

The current that cannot be pushed #

The meridional lines circle the ring, so by Ampère’s law there is a current flowing through it, along the ring, round the axis. In the drawn mode that current at the ring is 89 per cent of the largest current anywhere on the equator. It is not a small current in an unimportant place. It is a large part of what makes the field.

A current in a conductor has to be pushed. Ohm’s law for a moving conductor says the current density is the conductivity times the electric field felt by the moving material, which is the electric field in the laboratory plus the flow’s velocity crossed with the magnetic field:

$$\mathbf{J} = \sigma\left(\mathbf{E} + \mathbf{v}\times\mathbf{B}\right).$$

Now ask what can push current round the axis at the O-ring, and take the two terms in turn.

The electric field cannot. If the field is steady, the induced electric field vanishes — an electric field that circulates exists only while a magnetic field is changing — so the integral of the electric field round any closed loop is zero, and in particular round the ring. Whatever the electric field does locally, it gives no net push round the ring.

The flow cannot either. On the ring the meridional field is zero, so the whole magnetic field there points round the axis, along the ring. A velocity crossed with a vector pointing along the ring gives a vector at right angles to the ring, never along it. However the iron moves at that place — across the ring, along it, at any speed — the term $\mathbf{v}\times\mathbf{B}$ has no component that drives current round the axis.

So at the O-ring the current is required and nothing drives it. In a conductor with any resistance at all, a current with nothing driving it decays, and the field it supports decays with it. A steady symmetric field is impossible.

The argument is short enough that its omissions are worth listing, because they are what makes it strong. It says nothing about the flow. The iron can be moving in any pattern at all, symmetric or not; the contradiction is between the field’s shape and Ohm’s law, at one ring. It says nothing about how strong the resistance is, provided it is not zero. And it says nothing about how big the conductor is. A perfect conductor escapes it, and only because a perfect conductor needs no push to keep a current going — but a perfect conductor cannot make a field either; it can only keep the one it started with.

How thin the failure is #

The argument locates the failure on a single ring. It is fair to ask how much that matters, since real fluids are not exactly steady and the conductor round the ring might be supplied from nearby. The figure below puts a number on it.

At each point along the equator, divide the current the field needs by the meridional field available there. The result is the flow speed that would be needed, if the flow crossed the field in the most effective direction, to drive that current — the speed at which $\sigma v B$ equals $J$. It comes out in units of the diffusion speed $\eta/R$, where $\eta = 1/\mu_0\sigma$ is the magnetic diffusivity and $R$ the radius.

Near the centre the requirement is tiny: half a diffusion speed at a tenth of the radius. It rises through ten and a hundred as the ring approaches and then goes to infinity at the ring itself, because the field it divides by goes to zero there. For the Earth’s core the diffusion speed is about $3 \times 10^{-7}$ metres a second — the field leaks one core radius in roughly four hundred thousand years — and the flows inferred from the way the field changes at the surface are about half a millimetre a second, some seventeen hundred times faster. Those flows are more than enough almost everywhere. They fall short in a strip only 5.6 kilometres wide.

That is the counter-intuitive shape of the result. The symmetric dynamo does not fail by being too weak overall. It fails in a sliver, and the sliver is enough, because the field lines that pass through it are the innermost loops of the whole structure and resistance eats them from there outward. A design that worked everywhere but in a ring a few kilometres across would lose its field from the ring. The proof has to be done at a point because the failure is at a point; the quantitative version only shows how little room there is for anything to rescue it.

The same conclusion survives without the assumption that the field is steady. George Backus showed in 1957, and others extended it in the following decades, that the meridional part of a symmetric field obeys an equation in which the flow can carry flux about but nothing can create it, so the meridional field decays at least as fast as resistance alone would make it. The toroidal part can be made from the meridional part — that is the next section — but not the other way round.

What winding does, and what it cannot do #

Every differentially rotating conductor has a mechanism that looks as though it ought to escape all this. Wrap a meridional field line round a cylinder that turns faster at the equator than at the poles, as the Sun does, and the line is drawn out into a long spiral running mostly round the axis. A weak meridional field becomes a strong toroidal one. It is the same stretching that the stress of a field line turns into a force, and in the Sun it is what builds the bands of toroidal field that rise through the surface as sunspot pairs.

But the source of the winding is the meridional field, and nothing in the winding puts meridional field back. A symmetric flow can turn meridional into toroidal and has no way to do the reverse, because doing the reverse would need exactly the current along the O-ring that the previous section showed cannot be driven.

The figure reduces the whole problem to two numbers, the strength of the meridional field and of the toroidal one, each leaking away at its own decay rate. With only symmetric motion the toroidal field is fed by the meridional one through the shear and the meridional field is fed by nothing. The toroidal field rises for a while, peaks, and then decays at the same rate as its source, which is decaying on its own.

The way out was found by Eugene Parker in 1955. Rising and sinking blobs of conducting fluid in a rotating body are twisted by the Coriolis force as they move, and a twisted blob carrying a toroidal field line twists a small loop of it into the meridional plane. Many such loops, all twisted the same way because the rotation sets the sense, add up to a meridional field. Max Steenbeck, Fritz Krause and Karl-Heinz Rädler turned the idea into a theory in 1966 by averaging over the small motions and keeping their net effect, which they called the α effect: a contribution to the averaged electromotive force proportional to the averaged field itself.

The essential point is that the blobs are not symmetric about the axis. Each one is a small, local, lopsided motion. Their average can be symmetric, and the averaged field can be nearly so, but the thing doing the work is the departure from symmetry, which is exactly what Cowling’s argument forbids leaving out. In the caricature, once the product of the α effect and the shear exceeds one, the two fields feed each other and grow together. The real problem then has to stop the growth, which the caricature cannot do, and the stopping is where the field’s strength is decided.

What a field forgets #

The decay times that set the whole scale come from the same diffusion that makes a magnet fall slowly through a copper pipe and sets how far a field gets into metal. The slowest mode of a conducting sphere decays in a time $\mu_0\sigma R^2/\pi^2$, quadratic in the size, and the range it covers is enormous.

For the Earth’s core, every age that matters is longer than the decay time by a large factor. The field recorded in the oldest rocks is ninety thousand decay times old. Mercury, smaller and with a less certain conductivity, would forget its field faster still, and it has one, weak but present, found by Mariner 10 in 1974 and mapped by MESSENGER. Jupiter’s interior is large enough that a field could in principle survive from the planet’s formation, but its field is strong, changes on human timescales and has a structure only a working dynamo explains. Every planetary field that has been examined closely is being made now.

The Sun is the case that started the question. Its interior conducts well enough and is large enough that a field from its birth might linger, but its surface field reverses every eleven years, which no fossil could do, and the reversal is the clearest evidence anywhere that a dynamo is running.

A field that is never quite symmetric #

If Cowling is right, the Earth’s field cannot be exactly symmetric about the axis the core turns round. The surface field, measured by satellites and observatories and summarised every five years in the International Geomagnetic Reference Field, can be checked against that.

The dipole, which carries almost all the power at the surface, is tilted 9.4° from the rotation axis, and the tilt puts 2.7 per cent of its power into the asymmetric part. The smaller-scale parts are mostly asymmetric. The field is dominated by something nearly symmetric, with a lopsided remainder that is not small in proportion.

That is the pattern the theorem predicts and no more. It does not say that the asymmetric part must be large; it says only that it cannot be zero. A dynamo can make a field that looks symmetric to a compass while depending on asymmetric motion underneath, and every numerical model of the core that reproduces the Earth’s field does exactly that — columns of convecting iron aligned with the rotation axis, each one lopsided, together producing a dipole close to the axis.

The two laboratory dynamos that first worked, in Riga and Karlsruhe in 1999 and 2000, were built with the theorem in mind. The Riga experiment drove liquid sodium in a helical flow down a pipe and back up an outer jacket, and the field it generated spiralled round the pipe and drifted along it — a field of the kind the theorem allows, with no symmetry about the pipe’s axis. The Karlsruhe experiment pumped sodium through an array of fifty-two helical channels, each a small spiralling flow, arranged so that their averaged effect was a large-scale α effect of the kind Parker had imagined. Both made fields that sustained themselves, and both did it by being lopsided on purpose.

Where the argument stops #

Cowling’s theorem is one of a family of anti-dynamo results, each of which rules out a symmetry. A field that is the same along one straight direction cannot be kept up, nor can any field by a flow confined to planes. Each was found by the same route: assume a symmetry, find where the field must vanish, and show that the push the field needs there cannot be supplied. They are prohibitions, and like the prohibition a perfect conductor obeys, what is interesting about them is what has to break for the forbidden thing to happen anyway.

What none of the theorems does is say that a dynamo is possible. That took until 1958, when George Backus and Anthony Herzenberg separately found flows in a sphere that provably sustain a field — Herzenberg’s with two spinning spheres embedded in a conductor, deliberately and visibly lopsided. Ever since, the existence of dynamos has been a matter of proof for particular flows and of computation for realistic ones.

What the drawings cannot show #

The field lines and the speed curve are drawn for one mode of one sphere, chosen because it is an exact solution. A real core’s field has many modes at once, a solid inner core in the middle, an outer boundary that is not a perfect insulator, and a flow that changes over centuries. The O-ring argument applies to every symmetric field whatever its details, but the numbers attached to it — 0.873 of the radius, 89 per cent, 5.6 kilometres — belong to the drawn mode and to a diffusivity of one square metre a second, which is uncertain by a factor of two or more. The conductivity of iron at core pressures and temperatures has been revised substantially within the last fifteen years, and the decay time moves with it.

The caricature in the fifth figure is not a model of anything. It shows the logic — symmetric shear feeds one field from the other and never back — but its α effect is a single number standing in for the averaged twisting of countless eddies, and it contains nothing that limits growth, so it grows for ever. Real dynamos saturate when the field becomes strong enough to resist the motion, and how they do it, and what field strength they settle at, is decided by physics the two numbers do not have.

And the figure of the Earth’s field shows the field at the surface. The field at the top of the core, two thousand nine hundred kilometres down, is far more asymmetric than the surface field suggests, because the small-scale parts fall off more steeply with distance and the surface sees mostly the dipole.

The domain of the argument is any conductor with non-zero resistance and a magnetic field exactly symmetric about some axis. Inside it, no flow sustains the field. Outside it — any departure from symmetry, however small, in the field — the theorem is silent, and whether a dynamo works becomes a question about the particular flow.

Still open: why the dipole is so nearly aligned #

The geomagnetic dipole sits within about ten degrees of the rotation axis now, and averaged over ten thousand years or more it lines up with the axis to within a few degrees — the assumption that lets palaeomagnetists read ancient latitudes from the magnetisation of rocks. That alignment is what rotation should produce, because the Coriolis force organises the convection into columns parallel to the spin. But it is also as close to symmetry as the theorem permits a working dynamo to come, and the field reverses its polarity at irregular intervals, hundreds of thousands of years apart on average, passing through states in which the dipole is weak and tilted far over. Whether the near-symmetry and the reversals are two faces of one property of rotating convection, and what decides how close to the forbidden symmetric state a planet’s field can sit, are questions numerical models are only beginning to reach at realistic parameters.

The theorem itself is settled, and it is short. A field symmetric about an axis must have a ring on which its meridional field vanishes, and on that ring the current the field needs — 89 per cent of its largest value, in the slowest mode of a sphere — has nothing to drive it, because neither a steady electric field nor any flow crossing a field that points along the ring can push current along it. The Earth’s core forgets its field in forty thousand years and has kept one for three and a half thousand million, and it can do that only because the flow that keeps it is never, anywhere, quite symmetric.