One step of memory kept at the poles

Sunspot numbers have been recorded continuously since 1749 and sporadically for a century before that. It is one of the longest quantitative records in any science, it plainly oscillates with a period near eleven years, and until recently nobody could predict the next cycle’s size at all. The period itself is a reliable clock; the amplitude was not.

The failure is worth stating precisely, because it is what makes the eventual success interesting. Cycles vary in amplitude by a factor of three or more. The obvious predictor — this cycle’s size — is almost useless for the next one. So is the average of the last few. So is any autoregressive model fitted to the series, which is the usual test of whether a time series carries its own future.

A series with no memory of its own #

The absence of persistence is the first substantive fact, and it rules out a large family of explanations.

If the solar cycle were an oscillator with inertia — a pendulum, a resonant cavity, anything storing energy in a form that carries over — then a large cycle would tend to be followed by a large one, and the series would have positive autocorrelation at lag one. It does not, to a good approximation.

If it were a random process with no memory at all, nothing would predict anything, and the record would be indistinguishable from noise with a preferred timescale. That is not right either: the period is far more regular than the amplitude, which is the signature of a system with a clock and a weak amplitude control rather than of noise.

What the record looks like is a process with a short memory — one step — plus a substantial random component. And a one-step memory means there is a state variable, carried from one cycle to the next, that is not the sunspot number.

There is a hint in the record itself about where to look, and it was noticed long before it was explained. Cycles that rise quickly to maximum tend to be large ones, and cycles that rise slowly tend to be small — the Waldmeier effect, catalogued in the nineteen-thirties. A relation between the shape of a cycle and its size is not a prediction, since the shape is only known once the cycle is under way, but it says that the amplitude is set early rather than accumulated, which points at an initial condition.

The other hint is geomagnetic. The disturbance level at the Earth during a solar minimum correlates with the next maximum, a relation used as a forecasting tool since the nineteen-seventies with no mechanism attached to it. What the mechanism turned out to be is that the geomagnetic index at minimum is partly a measure of the Sun’s large-scale open field, which is largely the polar field — so the empirical precursor was measuring the state variable indirectly all along.

What the state variable is #

The candidate has been available since Horace Babcock proposed it in 1961 and Robert Leighton made it quantitative in 1969, and it is the Sun’s poloidal field — the large-scale field running from pole to pole, of a few gauss, which reverses sign at each maximum.

The picture runs in two steps. The interior’s differential rotation takes a poloidal field and winds it up into a strong toroidal one, buried below the convection zone. Buoyant sections of that toroidal field rise and erupt through the surface as active regions.

The second step is the one that closes the loop, and it is the observation Babcock built on. Active regions are not aligned east–west: the leading spot sits systematically closer to the equator than the following one, by an angle that grows with latitude. That is Joy’s law, measured in 1919, and its cause is the Coriolis force acting on a rising flux tube. The tube itself is a concentration of field strong enough that its own pressure evacuates the gas inside it, which is why it is buoyant and why it rises in the first place.

Because the pair is tilted, the following polarity is on average further from the equator. As the region decays, its flux is dispersed by supergranular motion and carried poleward by the Sun’s meridional circulation — and the following polarity, being poleward, arrives at the pole preferentially. Enough regions doing this cancels the existing polar field and rebuilds it with the opposite sign.

So the poloidal field at the end of a cycle is built from the surface decay of that cycle’s own active regions, weighted by their tilt. It is the seed for the next cycle’s winding, and its strength is the state variable.

The mechanism has an unusual character worth pausing on: the crucial step happens at the surface, in full view, rather than in the opaque interior. Every ingredient — the tilt angles, the decay, the meridional flow, the resulting polar field — is directly observable, and has been observed. Simulations of surface flux transport, given the actual observed active regions of a cycle as input, reproduce the observed polar field evolution well enough that the model is used to fill gaps in the magnetograph record.

That is a strong position for a dynamo model to be in. The interior half is inferred and contested; the surface half is measured and largely settled, and the surface half is the one carrying the memory.

Why the prediction works and why it is only one step #

The chain is now explicit: polar field at minimum, wound into toroidal field, erupted as spots, dispersed to rebuild the polar field. Each cycle’s output is the next cycle’s input, and nothing older survives.

That is precisely a one-step memory, and it is why the polar field predicts the next amplitude and the amplitude does not predict the one after. The information is destroyed at each step — the toroidal field is entirely consumed in producing the spots, and the new polar field depends on the tilts and on where the regions happened to emerge, both of which have a large random component.

The randomness is not incidental. The scatter in Joy’s law is large: individual active regions vary in tilt by tens of degrees about the mean, and a single very large region emerging at an unusual tilt can measurably change the resulting polar field. So the cycle’s amplitude is genuinely stochastic, seeded by a small number of large events, and the predictability is bounded not by measurement but by the physics.

That bound is roughly one cycle. Predicting two cycles ahead requires predicting the tilts of active regions that have not emerged, which is not possible even in principle.

The stochastic element also explains a feature of the record that looks like a puzzle. Cycle amplitudes show no long-term trend, no period-doubling, no evidence of the deterministic chaos that a nonlinear oscillator would produce — they look like a mean with fluctuations. A process whose amplitude is reset each cycle by a noisy input does exactly that, and searches for chaotic structure in the sunspot record, of which there have been many, have found nothing convincing.

There is a nonlinearity in there somewhere, because the amplitude does not run away. The leading candidate is a tilt-quenching: a stronger cycle produces flux tubes rising faster, less deflected by the Coriolis force, hence less tilted, hence contributing less to the next polar field. That is a negative feedback with the right sign and it is measurable in the tilt statistics, and the measurement is at the edge of what the data support.

The prediction that was made and tested #

The test came with cycle 24, and it is one of the cleaner examples of a physical prediction in solar physics.

The polar field measured during the minimum of 2008 was the weakest in the era of systematic magnetograph observation, roughly forty per cent below the previous minimum’s. The flux-transport account therefore predicted a weak cycle 24 — around half the amplitude of cycle 23.

That was an unpopular prediction. A panel convened to forecast the cycle split roughly in half, with the other camp predicting a strong cycle on the basis of methods extrapolating from the record itself or from geomagnetic indices.

Cycle 24 peaked at about half of cycle 23, the weakest in a century. The polar-field precursor was right and the extrapolations were wrong, and the argument has largely settled since.

It is worth being fair about how strong a test that was. One cycle is one data point, several methods predicted low, and a prediction made from a physical model that also has free parameters is not the same as a parameter-free one. What makes it more than anecdote is that the prediction was made in advance, publicly, from a measured quantity, against a majority view — and that the same method applied retrospectively to the cycles for which polar field data exist reproduces their amplitudes.

The practical stakes are not negligible. A cycle’s amplitude sets the ultraviolet output that heats and expands the upper atmosphere, and therefore the drag on everything in low orbit; it sets the frequency of the eruptions that drive geomagnetic storms; and it sets the cosmic-ray flux, which is modulated downward when solar activity is high — a modulation that works because an energetic particle’s path through the heliosphere is a random walk in a magnetised wind. The 2020 minimum’s polar field was slightly stronger, and the corresponding prediction was for a cycle 25 modestly larger than 24 — which is what has happened, though the margin is not large enough to count as a strong test.

What is actually measured, and how badly #

The polar field is a small quantity measured at the worst possible place on the Sun.

The poles are seen edge-on from the ecliptic, so the measurement is made at extreme foreshortening, through the maximum path length of the atmosphere, on a field that is nearly transverse to the line of sight — which is the geometry the Zeeman effect handles worst and the Hanle effect handles best. The consequences are real. Different observatories’ polar field measurements disagree by tens of per cent, the disagreement is systematic rather than random, and reconciling them is an ongoing project — much the same difficulty any small signal measured against a computed baseline runs into. Substitutes are used — the number of polar faculae, counted on white-light images, correlates well with the field and extends the record back further.

There is also an interval of only about six months per year when each pole is tilted toward the Earth enough to be measured at all, so the “polar field at minimum” is an average over a window rather than a snapshot.

A spacecraft in a solar polar orbit would resolve most of this at a stroke, which is one of the standing arguments for such a mission. Reaching a solar polar orbit is expensive — it requires either an enormous plane change or a gravity assist at Jupiter — which is why so few have flown.

What a one-step memory says about the dynamo #

The predictive success is useful and the structural conclusion is more interesting: the surface flux transport is not a by-product of the dynamo, it is a link in it.

That is a specific claim and it has a specific rival. In an interface dynamo the poloidal field is regenerated in the interior, by the helicity of convective motions acting on the toroidal field, with no surface involvement at all. Such a dynamo would have no reason for the polar field to predict anything, because the polar field would be a symptom rather than a cause. The prediction’s success is therefore evidence about the mechanism rather than only about the next eleven years, and it is one of the few pieces of evidence available. The interior is opaque to everything except sound waves, and what the sound waves say about the rotation constrains the winding but not the regeneration. What they do fix precisely is the shear layer at the base of the convection zone, and the fact that it has not spread is itself usually read as evidence that a field is confining it.

There is a caveat that keeps the question open. A dynamo can have both mechanisms operating, and if the interior one dominates in some regimes the correlation would hold in ordinary cycles and fail in extreme ones. Since the extreme ones are what matter — the Maunder minimum, when spots essentially disappeared for seventy years — the regime where the model is least tested is the regime of most interest.

The circulation that carries it #

One ingredient in the chain has been mentioned and not examined, and it is the one that sets the timescale.

The Sun’s meridional circulation is a slow poleward flow at the surface, about ten to twenty metres a second, with a return flow somewhere below. Ten metres a second carries material from mid-latitudes to the pole in about a decade, which is the right order for the cycle, and that coincidence is not lost on anyone. Where and how the flow returns is the largest open question in the model. A single deep cell returning at the base of the convection zone gives a transport time of a decade or two and a cycle period tied to the flow speed; multiple stacked cells give something else. Helioseismic inversions of the deep flow are at the limit of what the data support, and different groups have published incompatible answers over the last fifteen years.

The stakes are that the period’s origin changes. If the flow speed sets the period, then a star rotating differently should have a different cycle length in a calculable way; if the period comes from the dynamo’s own growth rates, it should not. The stellar cycle data are not yet good enough to decide, which is a recurring shape in this subject.

Grand minima, where the model is least tested #

The record contains an interval, roughly 1645 to 1715, during which almost no sunspots were seen. It is called the Maunder minimum, it is corroborated by the absence of aurorae in the same period and by the carbon-14 record in tree rings, and it is not explained.

A flux-transport dynamo with a stochastic tilt can reproduce grand minima, in the sense that a run of unusually weak polar fields can drive the cycle below the threshold at which spots erupt at all. Whether it then recovers depends on whether some residual poloidal field survives to be wound up, and different models differ. The observation that would settle it is not available for the Maunder minimum and might be for the next one: whether the cycle continues underneath, with the toroidal field simply failing to reach the surface, or stops entirely. Helioseismic frequency shifts continued through the deep minimum of 2008 in a way that suggested the cycle was still running, which is a hint at the answer in a very mild case.

The wider version of the question is what other stars do, and the answer is that they do something similar and not identical. Long-term monitoring of chromospheric activity in solar-type stars finds cyclic behaviour in roughly a third, irregular variability in another third, and flat activity in the rest — with the flat ones plausibly being stars in their own grand minima. That is the closest thing to a controlled experiment available. The Sun’s own record is one realisation of a stochastic process, and a hundred other stars are a hundred more.

There is a limitation to that comparison worth naming, because it decides how much the stellar data can settle. What is monitored on other stars is chromospheric emission, which is a proxy for surface magnetic flux rather than a measurement of a polar field. So the stellar samples can report whether a star cycles and at what period, and cannot report the quantity that does the predicting on the Sun.

Measuring a stellar polar field would need Zeeman–Doppler imaging — reconstructing a surface field map from the way circular polarisation in a line varies through a rotation — and that technique is available only for stars rotating fast enough to smear their lines usefully, which excludes stars like the Sun. So the one measurement that would test the mechanism elsewhere is precisely the one the mechanism’s own slow rotators forbid.

What the stellar data do supply is the boundary condition. Activity, cycle period and cycle regularity all vary with rotation, and rotation varies with age through magnetic braking, so a survey across stars is a survey across dynamo regimes. Where the Sun sits in that survey — near the transition from regular cycling to irregular activity — is itself a result, and an uncomfortable one for anyone treating the solar cycle as the typical case. A star chosen at random with the Sun’s mass is about as likely to show irregular activity as a clean eleven-year cycle, and the reasons for the difference are not known.

The obvious candidate is rotation rate, since that is what the dynamo runs on, and it is partly right: the fastest rotators are uniformly irregular and the slowest are uniformly inactive. The trouble is the middle, where the Sun sits, and where stars of nearly the same rotation period do quite different things. Metallicity, convection zone depth and binarity have all been proposed and none accounts for the spread. The samples are still small enough that the spread may partly be measurement: a cycle takes decades to detect, and the longest stellar activity records are sixty years old. Six cycles is not a sample from which regularity can be established, and the Sun itself would have looked irregular if observed for sixty years starting in 1645. The record’s length is the binding constraint on almost every statement in this paragraph, and it lengthens at one year per year. What that comparison has established is that cyclic behaviour is a property of stars rotating at roughly the Sun’s rate, that faster rotators are more active and less regular, and that the Sun sits close enough to the boundary that its own behaviour may not be typical of anything. It is a slightly uncomfortable conclusion for a subject whose only well-measured example is the Sun, and it is the reason stellar activity surveys are treated as part of solar physics rather than as a separate field.