A dynamo that did not need the shear layer

Inside the Sun, at seven-tenths of its radius, a convection zone that rotates faster at its equator than at its poles sits on a radiative interior that turns as one rigid piece. Between them is a layer about four per cent of the radius thick where the rotation rate changes from one to the other, thinner than any hydrodynamic argument says it should be. The layer is called the tachocline, and for thirty years it has been the favoured site of the solar dynamo.

The reasoning is attractive. A dynamo needs two things: shear, to stretch the poloidal field into a strong toroidal one, and something to regenerate the poloidal field from the toroidal. The tachocline has the strongest shear in the Sun. It lies just below the convection zone, where magnetic flux could be stored against buoyancy long enough to be amplified to the tens of kilogauss that the tilts of sunspot pairs seem to require. And the latitudes where spots first appear and the speed at which they migrate towards the equator can be fitted by dynamos rooted there.

The case for the layer is stronger than convenience. A magnetic flux tube in the convection zone is lighter than its surroundings, because its magnetic pressure lets it hold the same total pressure with less gas, and it rises. In the body of the convection zone a tube of the strength needed to make a sunspot would rise to the surface in weeks to months, far too quickly to be wound up over a cycle of eleven years. Just below the convection zone, in the slightly subadiabatic gas of the tachocline and the overshoot layer beneath it, buoyancy is opposed, and a tube can be held long enough to be stretched. Simulations of tubes rising from there reproduce the tilts and latitudes of emerging spot pairs if they start with fields of tens of kilogauss — ten times the equipartition strength of the convection near the base, which is itself an argument that the field was amplified somewhere convection could not shred it.

The tachocline also offers a clock. A dynamo wave travelling along the layer, its direction set by the sign of the shear and of the helicity, can be made to migrate towards the equator at the speed the butterfly diagram shows. Dynamos placed entirely in the convection zone, where the rotation varies mainly with latitude rather than depth, tended to produce waves that went the wrong way. For a while the layer seemed not merely sufficient but necessary.

What the Sun cannot do is switch the tachocline off and see whether the cycle stops. Other stars can. Below about 0.35 solar masses a main-sequence star is convective all the way to its centre. It has no radiative core, so it has no boundary between a rigid core and a differentially rotating envelope, and no tachocline. If a dynamo needs that layer, these stars should not have one — or should have a weaker or different one. They are among the most magnetically active stars known.

One relation for two kinds of star #

Magnetic activity in a cool star is measured most cleanly by its X-rays, which come from a corona heated by the magnetic field, expressed as a fraction of the star’s total luminosity so that stars of different sizes can be compared. That ratio depends on rotation, because rotation drives the dynamo. It depends on it through the Rossby number: the rotation period divided by the time a convective cell takes to turn over. A low Rossby number means rotation acts strongly on the convection, twisting it and organising it, and the dynamo is strong.

The relation drawn is the one measured for 824 stars by Wright and colleagues in 2011: a flat, saturated branch at an X-ray ratio of about one part in a thousand for rapid rotators, and a steep decline, as roughly the inverse cube of the Rossby number, for slower ones. The sample in the figure is synthetic — drawn from that relation with a scatter of 0.25 dex — so that what follows can ask what such a sample can and cannot show. The measured relation looks like it.

The question is where the fully convective stars fall. In the saturated regime they were known for decades to sit on the same plateau as Sun-like stars. The unsaturated regime was harder, for a reason the figures below make clear, and the decisive result came in 2016, when X-ray observations of four slowly rotating fully convective stars placed all four on the same declining branch as the stars with tachoclines. A larger sample two years later found the same thing. Stars with no shear layer have a dynamo that depends on rotation and convection in the same way as stars with one.

That is the cleanest argument available that a tachocline is not necessary for a dynamo. Before relying on it, though, it is worth asking what the agreement is made of.

The agreement is made by the turnover time #

The relation is in Rossby number, and the Rossby number contains the convective turnover time, which nobody measures. Plotted against the rotation period alone, the same stars do not agree at all.

A fully convective star of a fifth of a solar mass has a convective turnover time close to ninety days, against about fifteen for the Sun. Its dynamo saturates up to rotation periods six times longer, and at a given period it is far more active. The two kinds of star are put on one line by dividing each period by its own turnover time; without that division, they form two separate sequences.

So the agreement is exactly as good as the turnover times are. In a mixing-length picture of convection the turnover time is the size of the largest eddies divided by their speed, and both can be computed from a stellar model — but the result depends on where in the convection zone it is evaluated, and on the assumptions of mixing-length theory, which are weakest precisely in the deep, dense, slowly churning interiors of M dwarfs. What the published relations use instead is an empirical turnover time: one chosen, as a smooth function of mass or colour, so that stars of every mass fall on the same activity relation.

That is a legitimate procedure, introduced with the Rossby number itself in the 1980s, and it has a property that matters here. A turnover time calibrated from activity will make any group of stars lie on the relation, whatever their dynamo is doing.

What a calibration can absorb #

The way to see the danger is to simulate a world in which the fully convective stars really do have a weaker dynamo, and calibrate it.

In the first universe both kinds of star share one dynamo, and the turnover times solved from their activity follow the empirical curve smoothly from 1.2 solar masses down to 0.1. In the second, fully convective stars are three times less active at a given Rossby number — the kind of difference a dynamo deprived of its shear layer might show. Calibrating the turnover times from that universe’s activity does not reveal the difference as a misfit. It absorbs it: the fully convective bins come out with turnover times about 0.16 dex shorter than the curve, and with those times every star lies on one relation.

So a single activity–Rossby relation for both kinds of star is evidence of a shared dynamo only together with something the relation itself cannot supply: a turnover time that is smooth through the fully convective boundary for independent reasons. The empirical calibrations are smooth there. There is no reason in the convection itself for a step at 0.35 solar masses — the convective envelope simply deepens until it reaches the centre — and the turnover times computed from models also rise smoothly through the boundary. The step the second universe needs is not seen. But the argument has a joint in it, and the joint is the turnover time.

The part of the relation that needs no turnover time #

There is a part of the evidence that has no such joint, and it is the less dramatic half.

Saturation is a plateau. Below a Rossby number of about a tenth, the X-ray ratio no longer rises with faster rotation; it stays near one part in a thousand whatever the period. A star’s position on the plateau therefore does not depend on its turnover time at all, provided only that it is on the plateau, which a fast enough rotation guarantees. The saturated level is a direct, calibration-free measurement of how strong a dynamo can get.

And it does not change at the fully convective boundary. The base of the convection zone moves inward from seven-tenths of the radius in the Sun to about four-tenths at 0.4 solar masses, and then, over a few hundredths of a solar mass, the radiative core shrinks to nothing. A dynamo whose strength was set by the shear at the base of the convection zone would have to change across that transition — to weaken, or to saturate at a different level, or to behave erratically while the shrinking core still had a tachocline of sorts. The saturated X-ray ratio of real stars continues straight through.

What saturation means physically is still debated: the filling of the whole surface with active regions, a limit on the dynamo, or a reorganisation of the field into geometries that hold less of the corona in closed loops. Each explanation places the limit in the convection zone or the corona. None needs a tachocline, and the fact that the limit is the same with and without one is the observation that says so.

Why the slow ones took so long #

The unsaturated branch is where a dynamo’s dependence on rotation actually shows, and for fully convective stars it is hard to reach.

A Sun-like star leaves saturation at a period of about two days and reaches a Rossby number of one at about two weeks; the Sun itself, at 25 days, sits at about 1.7 on this scale. A star of 0.15 solar masses needs a period of two weeks to leave saturation and more than a hundred days to reach the same Rossby number. Fully convective stars are born spinning fast and are braked slowly, because their winds carry away angular momentum at a rate that depends on a field that stays saturated for billions of years. Those that have slowed to periods of months are old, and they are faint in X-rays — a thousand times fainter than when they were saturated, and intrinsically faint stars to begin with.

Measuring a period of a hundred days is also hard. A star’s rotation shows as a slow modulation of its brightness by spots, and a modulation of a per cent over a hundred days has to be separated from instrumental drifts and from the spots’ own evolution. The periods of slowly rotating M dwarfs came from ground-based monitoring programmes that watched thousands of them for years, and only then could X-ray observations be aimed at the slow ones. That is why the four stars of 2016 were four, and why the result was news.

How many stars a slope needs #

Lying on the line and lying along it with the same slope are different claims.

Four stars with Rossby numbers spread over a factor of ten, each with a quarter-dex of scatter, measure a slope only to about ±0.65. They show that fully convective stars on the unsaturated branch are not far above or below the relation, which is what the 2016 result established; they cannot show that the slope is the same. Distinguishing the slope of −2.7 from one of −2.2 — a difference that would point to a different scaling of the dynamo with rotation — needs about sixteen stars at two standard deviations, and a measurement precise enough to test dynamo models needs several dozen.

The samples since have grown towards that size, and they find the fully convective stars on the same branch with a slope consistent with the Sun-like stars’ within its uncertainty. Some analyses find a modestly shallower slope across the whole relation than the original −2.7, which applies to both kinds of star alike. Neither result separates the fully convective stars from the rest.

What the dynamo is doing instead #

If the tachocline is not needed, something else generates the field in a fully convective star, and the candidates come from the convection itself. Rotating convection is helical — rising and falling cells are twisted by the Coriolis force in opposite senses in the two hemispheres — and helical flow regenerates field in both directions, poloidal from toroidal and toroidal from poloidal, without needing a separate shear layer. Such a distributed dynamo had been studied theoretically for decades, and three-dimensional simulations of fully convective stars produce strong, large-scale fields from it, with the differential rotation suppressed by the field itself.

The fields of the stars agree with that picture in one respect and complicate it in another. Maps of the surface fields of rapidly rotating M dwarfs, reconstructed from the polarisation of their spectral lines, show that many fully convective stars carry strong, simple, axisymmetric dipoles — far more ordered than the Sun’s field. But stars of nearly the same mass and rotation can instead carry weak, complex fields, as if two different dynamo states were available and a star could settle in either. That bistability is not predicted by the simplest models, and it suggests that the fully convective dynamo is a different solution from the Sun’s rather than the same one without its foundation.

Cycles exist too. Proxima Centauri, a fully convective star of about an eighth of a solar mass rotating once every 83 days, has an activity cycle of about seven years in its X-ray and optical emission, recorded over more than a decade. A cycle is the sign of a dynamo that reverses or oscillates, not merely one that maintains a field, and it was the property most often attributed to the tachocline’s shear.

What this does and does not settle for the Sun #

The fully convective stars show that a dynamo can be strong, can saturate at the same level and can scale with rotation in the same way without a tachocline. They do not show that the Sun’s dynamo works without one. The Sun has a tachocline, and the processes that might use it — storing flux, amplifying it to strong toroidal fields, setting the latitudes of emergence — may still be how the Sun’s own cycle runs. Two stars can reach the same activity by different routes, and the relation measures the outcome, not the route.

What changes is the burden of the argument. The tachocline was favoured partly because no other site seemed able to support a strong, cyclic, large-scale dynamo. Stars without one do all of that. The case for the tachocline in the Sun now has to rest on the Sun’s own evidence: on whether its polar field predicts the next cycle in the way a surface-flux-transport dynamo requires, and on whether helioseismology finds the cycle’s signature at the tachocline at all. The frequencies that map the Sun’s interior have found variations of rotation over the cycle in the convection zone, migrating with the activity belts; a clear cycle in the tachocline itself remains disputed.

What the figures leave out #

The samples are synthetic and drawn from a single relation, so the figures can show what a sample of a given size and scatter can establish, not what the real samples contain. The turnover time follows one empirical calibration by mass; others use colour, and differ by tens of per cent at the lowest masses. The base of the convection zone is drawn from approximate values for standard models, which differ in detail near the boundary, where the core’s disappearance depends on the equation of state and the opacities of cool, dense gas. And the X-ray ratio is a single number, while activity in M dwarfs comes also in flares, which dominate their X-ray output at some times and are averaged here into a mean.

Still open: where the solar dynamo lives #

The gyrochronology that reads ages from rotation assumes that stars of a given mass spin down along a single track, which needs the dynamo and its wind to depend on rotation in a predictable way. Fully convective stars spin down differently — they keep their fast rotation for billions of years, then brake abruptly — and whether that is because their dynamo changes state, because their field changes geometry, or because their interior rotates as one piece and has no reservoir to draw on is not settled.

For the Sun the question is narrower and harder. A dynamo that works in the convection zone of a fully convective star could work in the Sun’s convection zone too, with the tachocline as a refinement rather than the engine. Or the Sun could run a different solution altogether, one that uses the shear because it has it. Separating the two needs either a measurement of the toroidal field at the base of the convection zone, which helioseismology has not yet made, or a sample of stars just above and below 0.35 solar masses observed closely enough to see whether their cycles, rather than their mean activity, change at the boundary. Whether they do is not yet known.