One region that can change the next cycle

The Sun’s cycle of spots is the most familiar regularity in stellar astronomy, and its amplitude is notoriously irregular. Cycle after cycle, spot counts differ by a factor of two or more, with no pattern a careful statistician would trust. The account of what a magnetic clock read off a butterfly can do came with a modest qualification, and then a single useful predictor turned up: the strength of the polar field at the minimum of one cycle correlates well with the strength of the next. The poles are one step of memory, and only one.

That raises the next question, which that account mentioned and left at a sentence. If the polar field decides the next cycle, what decides the polar field? The answer in the Babcock–Leighton picture is the tilt of active regions. A bipolar region’s two patches of opposite magnetic polarity are not quite aligned with the equator, and when that region decays, surface flows carry its trailing part poleward further than its leading part. The sum of many such contributions reverses the old polar field and builds the new one. A cycle’s several thousand regions all add a small amount to a running total.

The picture sounds like an average, of the sort that would be stable and easy to calculate. It is nothing of the kind, for two reasons that can each be written down in a line. A region’s contribution has a very strong dependence on where it emerges, and its tilt is drawn from a distribution with more scatter than signal. Together they mean that a handful of regions matter out of proportion to their number, and that the next cycle’s strength has a random component that no future measurement of the present cycle could remove.

Most regions never reach the poles #

A region emerging at thirty degrees of latitude has a leading and a trailing patch, both in the same hemisphere. Diffusion spreads both of them, and the meridional flow takes both towards the pole. They arrive largely together, cancel each other out, and leave little net polar flux of either sign. A region that emerges close to the equator behaves quite differently. Its leading patch diffuses across the equator and cancels with the leading patches of the other hemisphere’s regions, which have the opposite polarity, so its trailing patch is left over and travels to the pole alone. That is the contribution that counts.

Flux-transport calculations show that this result is simple in form. The final contribution is close to a Gaussian in latitude, with a width of about nine degrees, set by the ratio of the rate at which flux diffuses to the speed of the meridional flow. A region’s initial contribution, on the other hand, depends on how widely separated its patches are and how far it is tilted. Taking the two together, the most effective regions lie at a latitude where tilt has begun to grow and the weight has not yet collapsed, at around ten degrees.

That means a large fraction of the regions in the butterfly diagram are nearly irrelevant to the future. Those at the top of the butterfly, so prominent early in each cycle, produce some of the largest spots ever photographed, and their long-term contribution to the next cycle is almost nothing. The spots that make the news need not be the ones that matter. What matters tends to be less conspicuous, and it comes at a time when the cycle seems to be running down.

A law with more scatter than signal #

The tilt has been known since Alfred Joy’s measurement from Mount Wilson plates, published with George Ellery Hale in 1919. Its origin is the Coriolis force on the flux tube as it rises through the convective zone. It rises with latitude, about half a degree of tilt for every degree of latitude near the equator.

The scatter is what has to be appreciated. A rising flux tube is shredded and rotated by convection, and it may emerge at almost any angle. Near the equator the mean tilt is close to zero, and the scatter stays large, so close to half the regions there are tilted against Joy’s law. These anti-Joy regions are not a rare exception, and they take away from the polar field rather than adding to it. So the region-level picture is one of a small mean rising out of a large random term, and that is the first ingredient in what follows.

Such scatter is also a measurement challenge. Tilts are read from magnetograms, and a tilt depends on what is taken as a single region, how a complex group is divided up, and the instant chosen, because regions rotate as they develop. Several historical catalogues, drawn from white-light sunspot drawings, measure a different quantity from magnetogram-based ones. A disagreement between catalogues over a cycle’s mean tilt has lasted for years, and it matters directly, because the polar field responds to that mean tilt.

When the next cycle is decided #

The combination of the latitude weighting and the butterfly’s slow equatorward drift has a clear consequence for timing.

Sunspot number peaks about four years into a cycle, when the number of regions is greatest but most of them are at mid latitudes. The polar field takes its shape well after that, from the fewer regions that emerge in the descending phase. In real cycles, the two factors work against each other to some degree, as the emergence rate is higher earlier and the weight higher later, so a full treatment gives the most important interval as around or just after maximum, consistent with the published finding that the strongest effects on the next cycle come from regions emerging around cycle maximum at low latitudes. In either version, the bulk of the outcome is not fixed until the cycle is past its peak.

This gives a sobering arithmetic for anyone hoping to forecast the following cycle. A prediction from the polar field cannot be made until the polar field is complete, and that is close to the minimum. Physics-based models can extend the lead time by estimating how the remaining regions will add up, but they cannot know the tilts or the sizes of regions that have not surfaced. It explains the gap between the two prediction records of cycles 24 and 25, in which the polar-field precursor did well and the many earlier predictions, made before the field was in place, scattered widely.

A rogue #

The final factor is a heavy tail in the sizes. A few active regions in each cycle are vastly larger than the typical one, and a region’s dipole contribution rises faster than its flux, since its polarity separation also grows with its size.

So a single region can be a large fraction of everything else put together, and it can do this with either sign, depending on its tilt. Melinda Nagy and collaborators, in 2017, tested the effect directly in a dynamo model, inserting one rogue region and following what happened. It could boost or suppress the amplitude of later cycles. In extreme cases a single region could shut the model’s dynamo down altogether and trigger a grand minimum. They found that the largest effect arose around cycle maximum at low latitudes, and that the region did not have to cross the equator. Jie Jiang and colleagues, examining the real cycle 23, found that the weak polar fields that preceded the weak cycle 24 were largely attributable to a limited number of large regions having the wrong tilts. So the real Sun appears to have done something of this kind within the last thirty years.

The term rogue is apt. These regions are not detectably different in origin. They simply fall at the far end of two long-tailed distributions at once, and they happen at the moment that matters. Nothing about one cycle predicts when or where the next will come.

Identical cycles, different poles #

The natural question at this point is what spread in the polar field one should expect from chance alone. That can be tested by running many cycles with exactly the same laws.

This is the second key figure. Two suns identical in every systematic respect will have polar fields that differ by something like twenty per cent as a matter of chance. Observed polar fields at cycle minima vary by a factor of about two from one cycle to another, and direct records cover only a few of them. So a substantial part of the variation in the amplitude of real cycles is plausibly plain sampling noise in a small number of influential events. The cycle does not need any exotic chaotic mechanism in the deep dynamo to be irregular, as the randomness is already there at the surface.

A further feedback might damp this. Strong cycles could have shallower mean tilts or more equatorward regions, and evidence has been presented for both, and that nonlinearity may be the reason the Sun’s cycle has stayed within limits rather than wandering away. Such feedback would reduce the scatter’s effect on the longer term, and it does not change the size of each step.

How far the memory reaches #

What does this do to the longest forecasts? Let each cycle keep a fraction of the previous polar field through whatever coupling links them, and add a fresh random term from the regions. The correlation then falls as a power of that fraction.

If one takes this reasoning seriously, then a forecast of the amplitude of a cycle two or more ahead can hardly exist. The actual observed memory is even shorter, about a single cycle, according to the record of the polar field as predictor. That finding is one of the main arguments for a short-memory dynamo, in which the turbulent diffusion wipes out earlier fields before they can build up.

Similar reasoning puts limits on the old idea that long-term variations can be anticipated. The Sun’s record of grand maxima and minima cannot be extrapolated easily, because the random term means a grand minimum might be reached by way of a sequence of bad luck. Grand minima appear in the isotope record with an irregular spacing that is consistent with a chance process, and a stochastic model with a threshold effect can produce episodes resembling the Maunder Minimum. If this is correct, the next grand minimum has no date, in the same sense that the next rogue region has no date.

Two hemispheres, one coin #

There is a natural experiment already running on the Sun, because it has two hemispheres whose regions are largely independent. If the polar field is a sum over a small number of influential events, the northern and southern totals should differ by chance, and they do. In cycle 24, the northern polar field reversed well over a year before the southern, and it hovered near zero for some time. The hemispheres’ sunspot numbers have also run out of step by a year or more in many cycles, and for several decades in the Maunder Minimum what few spots were seen lay almost entirely in the south.

This asymmetry has often been read as a sign of some deep structure in the dynamo, and there may be some. But a chance difference of about the size computed above is expected whatever else is going on. In the same way, a rogue region in one hemisphere changes the balance between the hemispheres directly, as Nagy’s team found in a later study, and a lopsided cycle can lead to a lopsided next one. The null hypothesis, then, is chance, and any claim for a systematic hemispheric cause has to show that it exceeds what a sample of a few influential regions would produce by accident. Very few such claims have tried.

Other stars offer a third kind of evidence. Decades of chromospheric monitoring at Mount Wilson and elsewhere have found stars whose activity varies smoothly in a cycle, others with irregular variation, and some with flat records that have been compared with the Sun in a grand minimum. Differential rotation inside a star and its overall activity should set the mean rate and size of its regions, and a star with a few very large regions, like the young active stars, should have far noisier polar fields than the present Sun. Young Sun-like stars do show a more erratic pattern, and their fields take a more complicated geometry at the same time. So the same statistical reasoning, carried over to other stars, suggests that the Sun’s fairly regular cycle is a feature of a relatively low activity, in which there are many small regions and few rogues.

Chance at the surface and order below #

This gives a striking combination. The dynamo as a whole produces an extremely regular eleven-year clock, whose phase has been kept for centuries, while its amplitude is left to chance. Those two properties seem at odds, but both are consistent with a cycle driven by the surface. The cycle’s period comes from systematic flows and diffusion, which do not vary much. Its amplitude comes from a sum over a few influential regions, and it varies. There is a corresponding lesson for how to read what spin and fields tell about other stars. A single star’s cycle is a noisy realisation, and much of the difference between two stars’ cycles in any short time span may have no physical meaning at all.

The surprise is in the unit of action. Most of stellar astrophysics works with means and with a star’s large-scale properties. Very few processes on a star’s surface can be attributed to single events, and it is unusual to say about a star of the Sun’s mass that one region, on one day, can change what the whole star does for the following decade. Yet a spot’s magnetic field is an affair of a few tens of thousands of kilometres, and it can do that.

What the picture cannot show #

The simulations take a very simplified form. Regions have a distribution of flux that is an exponential with a floor, migrating latitudes with a fixed spread, and a uniform rate of emergence, and the region’s dipole is a simple function of its flux. There is no flux transport here at all, only its result, a weight function assumed Gaussian. Cross-equatorial and hemispheric differences have been left out, along with active nests, in which regions emerge repeatedly in one place. The total is taken as the sum for one hemisphere, with no feedback from strong cycles to tilts or latitudes.

The rogue figure relies on a rough power law for the size scaling, and the true value differs from region to region. The horizon figure takes a fraction kept as an input. It therefore does not derive the short memory, and a lower or higher value would change the look of the figure substantially. The cycle shape in these figures also differs from the Sun’s, whose rate of emergence peaks a few years in. So the half-way point of the dipole would fall earlier in a realistic calculation, though it still comes after the peak in sunspot number.

Still open: whether tilt carries a signal or only noise #

The largest open question is whether the mean tilt changes systematically with cycle strength, since that is the most plausible way the cycle could regulate itself. If stronger cycles have smaller average tilts, because a stronger toroidal field rises more quickly and is rotated less, then a built-in brake exists. Different catalogues give conflicting results, for the reasons outlined above. A uniform analysis of the century-long records now digitised, with consistent definitions and with magnetograms for calibration, should reach an answer. There is a harder task as well, to observe a rising flux tube before it surfaces, by helioseismic imaging of the interior a day or two before emergence. Any tilt that could be seen beforehand would tell of a systematic cause. If it cannot be seen, the rogue regions will be as unpredictable as they look, and the next cycle’s strength will remain a matter of a few events, to be known only after they have happened.