The dynamo that reverses with nothing pushing it

The field no symmetric flow can keep ended with a prohibition. A conducting fluid moving symmetrically about an axis cannot keep up a magnetic field symmetric about the same axis, however fast it moves, because there is a ring inside such a field where the current it needs has nothing to drive it. The Earth’s core keeps its field only because its flow is never quite symmetric. The theorem says what cannot work. It does not say what can. A field left to itself in a conductor decays, as the field that cannot get out found when it treated the conductor as perfect and then let the resistance back in, and the flows that do work in a planetary core are three-dimensional, turbulent and so far reproduced only by some of the largest calculations in geophysics.

There is a much smaller machine that works, and it is old. Edward Bullard described it in 1955 as a model of what a dynamo has to do, stripped of everything else: a copper disc on an axle, a coil of wire wound round the axle beneath it, and two sliding contacts that send the disc’s current through the coil. It has two variables — the disc’s speed and the current — and its equations can be written on a postcard. It makes its own magnetic field from nothing more than a stray one. It cannot decide which way that field points. And in 1958 Tsuneji Rikitake joined two of them together and found that the pair reverses its field, over and over, at irregular intervals, with the motors that drive it held perfectly steady.

A generator that powers its own magnet #

The disc is Faraday’s, and the voltage it makes is the motional emf that the field that makes the other traced to a conductor moving through a field rather than to any field that changes. The rule that is two laws wearing one coat followed it as the case that breaks the flux rule: a conducting disc turning in a steady field drives a steady voltage between its axle and its rim, although no flux through any circuit is changing, because every free charge in the disc is moving across the field and feels the magnetic force along the radius. The voltage is the speed times the flux through the disc divided by $2\pi$. Connect a load between axle and rim and a steady current flows.

Bullard’s change was to make the load the magnet. The current from the rim goes down through a coil wound round the axle and back to the axle, and the coil’s field passes through the disc. Now the field the disc turns in is made by the current the disc drives. Write $M$ for the flux the coil puts through the disc per unit of current, divided by $2\pi$ — a mutual inductance — and the disc’s voltage is $M\Omega I$, where $\Omega$ is its angular speed and $I$ the current. The circuit has a resistance $R$, mostly in the disc and the sliding contacts, and a self-inductance $L$, which the circuit that fights its own change found resisting any change in the current. So

$$L\frac{dI}{dt} = (M\Omega - R)\,I.$$

That one line already holds the dynamo’s central fact. If $M\Omega$ is less than $R$ the bracket is negative and any current decays exponentially: the disc’s voltage cannot keep up with the circuit’s losses. If $M\Omega$ exceeds $R$ the bracket is positive and any current grows exponentially. The dividing line is the critical speed

$$\Omega_c = \frac{R}{M},$$

a resistance divided by an inductance, and above it the circuit amplifies whatever current it holds, with nothing else in it but copper.

The current does not grow forever, because it brakes the disc. A current crossing the disc’s radius in the coil’s field feels a magnetic force round the axle, opposing the rotation, with a torque $MI^2$. A motor drives the disc with a steady torque $G$, and some friction resists it. So the second equation is the disc’s own:

$$C\frac{d\Omega}{dt} = G - MI^2 - k\Omega,$$

with $C$ the disc’s moment of inertia and $k$ the friction. Two equations, and the figure at the top of this essay is their solution from rest. Speed is measured in units of $\Omega_c$, current in units of $\sqrt{G/M}$, time in units of the circuit’s $L/R$, and the motor’s strength by $g$, the speed it would reach with no current, also in units of $\Omega_c$.

At $g = 0.8$ the disc runs up to its free speed and stays there, below critical; the seed current of a millionth simply dies. At $g = 1.5$ and $2.5$ the disc passes the critical speed, and for a while nothing visible happens. The current is growing exponentially, at a rate set by how far the speed exceeds $\Omega_c$, but it is starting from a millionth. Eleven and a half factors of $e$ separate a millionth from a tenth, and at $g = 1.5$, with the disc running half as fast again as critical, the current gains one factor of $e$ every two time units, so it reaches a tenth only at $t \approx 29$. Then it arrives all at once. The current’s brake works the way the eddy currents in the magnet that falls slowly do, turning motion into ohmic heat through a force that grows with the current, and here it overshoots: the disc slows below critical and the current falls back, and the two ring against each other until they settle at a fixed point: the disc at exactly the critical speed, the current at $\sqrt{g-1}$, 0.71 and 1.22 in the two runs.

Exactly the critical speed #

The final speed is not an accident of the parameters. In a steady state $dI/dt = 0$ with $I$ not zero, so the bracket must vanish and $\Omega = R/M$ exactly, whatever the motor does. A stronger motor does not make the disc run faster. It makes the current larger, until the current’s brake absorbs the extra torque: in units of $G$, the brake is $I^2$, the friction at the critical speed is $1/g$ of the torque, and the balance gives $I^2 = g - 1$. Every joule the motor supplies beyond what friction takes goes into the circuit’s resistance as heat, and none into speed.

This is a thermostat, and the machine is its own. A disc running slightly above critical grows its current, which brakes it back; a disc slightly below critical loses current, which frees it to speed up. It is also the shape every self-sustaining dynamo has, the Earth’s included. The core’s flow does not run as fast as the heat escaping from it could drive it; it runs at whatever speed makes the field’s growth balance the field’s decay, and the energy beyond that goes into the field and through it into ohmic heat in the iron. The critical speed of the disc is the critical magnetic Reynolds number of a planetary dynamo, the ratio of flow speed times size to magnetic diffusivity above which the field grows, and a working dynamo sits on it.

The machine is not a curiosity of the laboratory. Every self-excited generator ever built works the same way. Werner von Siemens, Charles Wheatstone and Samuel Varley all described the principle in 1866 and 1867: let a generator’s own output energise its field magnets, and it will build up from the faint magnetism left in the iron of its poles. Electrical engineers still learn the two conditions that Bullard’s equation states — that the generator turn faster than a critical speed and that its field circuit’s resistance be below a critical resistance, two readings of $M\Omega > R$ — and they still learn that a machine whose iron has lost its remanent magnetism will not build up at all, and has to have its field “flashed” from a battery. The seed is not a mathematical convenience. Without one, the current is exactly zero and stays zero, because zero is always a solution.

A field of either sign #

The equations have a symmetry worth looking at directly. Change the sign of $I$ and nothing changes: the current’s growth rate depends on the speed, not on the current’s direction, and its brake depends on $I^2$. So any history with one sign of current has a mirror history with the other, and the faint second curves in the figure at the top are exactly that — seeds of $-10^{-6}$ reaching $-0.71$ and $-1.22$ along the reflections of the solid curves.

The plot of steady current against drive is a pitchfork. Below $g = 1$ there is one steady state, with no field. At $g = 1$ it loses its stability, and two new ones branch out of it, symmetric, differing only in the direction of the current and so in the direction of the field. Nothing in the machine’s construction prefers one. The coil’s winding sense and the disc’s direction of rotation decide whether the arrangement can excite at all — spun the wrong way, the coil’s field opposes the field it is fed by, $M$ is negative, and no speed is critical — but once the arrangement works, both polarities work equally well. The stray field present when the speed passes critical picks between them, and whatever picks it is outside the equations.

That is what a geomagnetic polarity is, in the simplest machine that has one. The Earth’s field has pointed in its present direction for about 780,000 years, and before that, at the Matuyama–Brunhes reversal, it pointed the other way; nothing about the core’s construction favours either. A dynamo’s laws are unchanged when its field is reversed — the magnetic force on a current depends on the field twice, once through the current the field induces and once through the field the force acts in — so every dynamo that can sustain a field can sustain its mirror image, and the question is never which polarity is possible but what, if anything, makes the field move from one to the other.

The disc answers: nothing does. Once it has chosen, a single disc keeps its field forever. Its fixed point is stable, and the ringing in the first figure dies away. A single disc cannot reverse.

Two discs that keep swapping #

Rikitake’s machine is two of Bullard’s discs, driven by equal motor torques, with their wiring crossed: the current from disc 1 runs through the coil under disc 2, and the current from disc 2 through the coil under disc 1. Each disc’s current now depends on the other’s field. Scaled in the same way, with $x$ and $y$ the two currents and $z$ a measure of the discs’ rotation, the equations are

$$\dot x = -\mu x + zy, \qquad \dot y = -\mu y + (z - A)x, \qquad \dot z = 1 - xy.$$

The constant $A$ is the difference between the two discs’ speeds, which the equations keep fixed, and $\mu$ measures the circuits’ resistance. With $\mu = 1$ and $A = 3.75$ the steady states are at currents of $\pm 2$ in disc 1 and $\pm\frac{1}{2}$ in disc 2, with $z = 4$; as with one disc, the field can point either way. But they hold nothing.

The current oscillates about one of its steady values, the swing grows, and after a number of swings that varies from one episode to the next it overshoots through zero and starts to oscillate about the other. In the first three hundred time units it reverses thirty-five times. Sometimes it holds one polarity for a dozen oscillations; sometimes it crosses back after one. There is no randomness anywhere in the model. The torque is constant, the circuits are fixed, and the equations are three lines of ordinary differential equations solved to within the accuracy of the arithmetic.

The figure of the orbit in the plane of the current and the rotation variable shows why. The two steady states are centres: a small disturbance near one of them circles it at a fixed period, 3.05 time units, neither growing nor shrinking in the linear approximation. The nonlinear terms make the circling drift outward, and once a turn is wide enough it passes close to $x = 0$, where the discs’ currents are small and the rotation is free to change. From there it can be thrown to either side.

The picture has two lobes joined at the middle, and anyone who has seen Edward Lorenz’s model of convection will recognise its shape: Lorenz wrote down his three equations in 1963, five years after Rikitake, and found the same thing in a fluid heated from below. Which lobe the orbit lands in after each crossing depends on exactly how wide its last turn was, which depends on the turn before, and so on back; a small difference in the starting state is amplified at every turn, as the error that doubles on a schedule found in a double pendulum, until the sequence of polarities is unpredictable beyond a few reversals. The machine is deterministic and its reversals are not forecastable.

Intervals that remember nothing #

A long run makes the irregularity quantitative.

The shortest interval is one turn round a steady state, about 3.1 time units: the orbit must go round at least once before it can cross. Beyond that the lengths spread out over a factor of ten, and their standard deviation, 8.9, is nearly as large as their mean, 10.1. The test of whether reversals keep any memory is to ask how much longer an interval lasts, on average, given that it has already lasted some time. For events that occur at random with a fixed rate, the answer does not depend on how long one has already waited. For the two-disc dynamo an interval that has lasted 5 goes on for a further 10.2 on average, one that has lasted 10 for 9.3, and one that has lasted 15 for 10.0. Through the bulk of the distribution the machine reverses as if by coin toss, though there is no coin in it. It is the same disorder that the calm that is the ghost of a cycle found in a system on the edge of chaos, long regular stretches broken at moments nothing in the record predicts, and it comes from the same place: a deterministic orbit passing again and again near a point where small differences decide everything.

The geomagnetic record has the same character. The sequence of reversals read from the magnetised stripes of the ocean floor, back through the last 160 million years, has intervals from a few tens of thousands of years to tens of millions, and for most of that time their lengths are close to what a random process at a fixed rate would give: Allan Cox pointed this out in 1968, and it was the first argument that reversals are a property of the dynamo rather than of something external hitting it. The longest interval, the Cretaceous Normal Superchron, lasted about 38 million years without a single reversal, and that is longer than the random model allows, a sign that the rate itself changes over tens of millions of years, presumably as the heat flowing out of the core across the mantle’s base changes.

What the discs leave out #

Rikitake’s machine is not a model of the core, and it was never meant as one. It has three variables; a core has as many as its turbulence has eddies. The discs’ conductors are wired, and the current must go where the wire takes it — which is how they escape Cowling’s theorem — the handedness that, in a fluid, only helical motion can supply, and whose magnetic counterpart the twist that outlives the turbulence followed as helicity — since a wire wound round an axle is a helix and has a handedness that no symmetric flow in a homogeneous fluid can supply. Its reversals are oscillations growing until they tip, and the geomagnetic field does not oscillate about its steady values in anything like that way; real reversals take a few thousand years, during which the dipole weakens to a tenth or less of its strength and the field becomes complicated, and then rebuild in the opposite direction.

What the discs carry over is the logic. A dynamo has a threshold, above which it amplifies any seed; it has a symmetry, which makes both polarities equally available; and a dynamo with more than one way of storing its energy can move between the polarities by its own dynamics, irregularly, with no outside cause. That chain survives into every more realistic model. Laboratory experiments with liquid sodium achieved self-excitation in Riga and Karlsruhe in 1999, and in 2007 the VKS experiment in Cadarache, in which two counter-rotating impellers stir sodium, produced a field that reversed at irregular intervals — a fluid dynamo doing what Rikitake’s discs had done on paper half a century earlier.

Still open: why the Earth reverses as rarely as it does #

The two-disc machine reverses every ten circuit times on average and the Earth reverses every few hundred thousand years, which is some ten free-decay times of its core and several hundred overturns of its convection. Numerical models of the core that run at Earth-like ratios of rotation to viscosity show reversals, but how often they reverse depends on how strongly the convection is driven, and models driven near the Earth’s estimated state tend to reverse either too often or not at all. Whether the Earth’s long intervals reflect a core that sits just below a threshold for frequent reversal, with rare excursions over it, or a mechanism that the models do not yet capture, such as the interaction of the core’s flow with the stably stratified layer that may sit at its top, is not settled. Neither is whether the present weakening of the dipole, by about nine per cent since 1840, is the start of anything.

The statement the discs make is short. A disc whose current runs through a coil wound round it makes its own field once it turns faster than $R/M$, settles at exactly that speed with a current of $\sqrt{g-1}$, and takes the direction of whatever stray field it started from, because its equations are unchanged when the current reverses; two such discs, each feeding the other’s coil and driven by a constant torque, reverse 1,187 times in 12,000 circuit times at intervals averaging 10.1 and nearly memoryless. A dynamo’s polarity is a choice its equations cannot make, and a dynamo with enough room inside it keeps making the choice again.