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A shear layer that should have spread

The Sun's convection zone turns differentially — its equator laps its poles about once every three months — and the radiative interior below turns as one rigid piece. Between them is a transition four per cent of the radius thick. Nothing in hydrodynamics keeps a velocity discontinuity that thin for four and a half billion years.

Assumes Internal rotation, Asteroseismology and Energy transport.

18 min read 10 figures Measuring the unreachableSmall perturbations, large consequences

The Sun does not rotate at one rate. Its equator turns once in about 25 days and its poles in about 34, and that difference has been known since sunspots were first tracked across the disc. What was not known until oscillation frequencies could be measured well enough is what happens below the surface.

The answer is that the differential rotation is a property of the part of the star that boils alone. It persists inwards, almost unchanged with depth, through the outer 29 per cent of the radius. Then it stops.

The Sun's internal rotation: differential above 0.693 R, rigid below, and the 15 nHz between two modes. Angular velocity against fractional radius, in nanohertz, from the inversion of hundreds of thousands of measured rotational splittings. Two features had to be discovered rather than deduced. The convection zone rotates differentially in latitude — 473 nanohertz at the equator, 330 at sixty degrees — and that latitude dependence persists all the way down through it, in surfaces that are very nearly radial rather than the cylinders a rotating convective fluid was expected to produce. And below about 0.693 of the radius the latitude dependence stops: the radiative interior turns as a single rigid body at 430 nanohertz, which is a remarkable thing for a fluid with no strength to do, and requires something — most likely a weak internal magnetic field — to be enforcing it. The two regimes are joined by a shear layer a few per cent of the radius thick, the tachocline, and it is where the solar magnetic cycle is generally thought to be generated, because it is the only place in the Sun with the shear a dynamo of that strength needs. The horizontal marks are what individual modes would report: the rotation averaged over the cavity each one occupies, weighted by the time the wave spends at each radius. A mode of degree 100 is trapped near the surface and reports 473 nanohertz; one of degree 1 passes through the deep interior and reports 458. The 15-nanohertz difference between them is the measurement, and splittings are measured to about a nanohertz — which is why the profile can be resolved at all. The kernels used here have the right support and the right sense; a real inversion uses computed eigenfunctions, and its resolution below 0.2 R is poor for the reason the previous figure gives.
Fig. 1 The Sun’s internal rotation, resolved with depth. The outer region rotates differentially — the same latitude dependence seen at the surface, running most of the way down without much change — and the region below turns as a single rigid body at an intermediate rate. The transition between them is measured at 0.693 of the radius and is a few per cent of the radius thick. The 15 nanohertz splitting between two modes marked on the figure is what the whole measurement rests on.

Why a rigid interior is strange

Consider what the radiative interior has been through. The Sun arrived on the main sequence turning perhaps ten times faster than it does now, and a magnetised wind has been braking it ever since. But the wind grips the surface. It has no direct hold on anything below the convection zone.

So the outer layers have been slowed by a factor of ten while the interior has been slowed by nothing at all — unless something transmitted the torque inwards. If nothing did, the interior would still be turning at its birth rate, ten times faster than the surface, and the rotation profile would show an enormous jump at the base of the convection zone.

It does not. The interior turns at very nearly the mean rate of the convection zone. Whatever couples them has been strong enough to hand the entire braking torque down through the radiative zone over four and a half billion years, and to do it well enough that no measurable residual gradient is left.

That is one of the two facts. The other is stranger.

The layer is too thin

The base of the convection zone is not a place where the rotation smoothly blends from one behaviour to the other over a comfortable distance. The transition happens in about four per cent of the solar radius — thirty thousand kilometres, out of seven hundred thousand.

A velocity shear in a fluid diffuses. The rate depends on the viscosity, and the microscopic viscosity of the solar interior is tiny, so the microscopic spreading is negligible. But a shear layer in a rotating, stratified fluid does not spread by viscosity alone: it spreads by a meridional circulation that the shear itself drives. That process — hydrodynamic spreading of a tachocline — has a well-defined timescale, and it is short. Left alone, the layer would have thickened to fill a substantial fraction of the radiative zone within the Sun’s lifetime.

It has not. Something confines it, and identifying that something is one of the standing problems of solar physics.

What the layer is asked to do

It is worth listing what the tachocline has been recruited for, because the list is long and the layer is thin.

It is where the solar dynamo is thought to generate its large-scale field, because it is the only place in the Sun where a strong shear coexists with a stable stratification able to hold a strong field down against its own buoyancy. A field generated in the convection zone proper would rise and escape within weeks; a field generated in the tachocline can be stored and amplified for years, which is what a cycle of eleven years requires.

It is the boundary that decides how much material is mixed downwards out of the convection zone, and therefore the surface abundance of every element destroyed at temperatures just below it.

And it is the interface across which the entire braking torque has to pass. The wind grips the surface; the convection zone communicates that grip downwards on a timescale of months; and everything below has to be reached through this layer.

Four different beginnings and one ending. Rotation period against age for four stars born turning at 0.3, 1, 3, 8 days, integrated under a braking law that goes as the cube of the rotation rate below a saturation period of 3.4 days and linearly above it. Both axes are logarithmic. The tracks span a factor of 25.7 at ten million years, 1.93 at six hundred million, and 1.09 at the age of the Sun — the initial condition is not merely diluted, it is erased, because the solution Ω = Ω₀(1 + 2KΩ₀²t)^(−1/2) tends to (2Kt)^(−1/2) with no Ω₀ left in it. The late slope measured off the drawn curve is 0.500 against the one half the law demands. The single constant K is fixed by one requirement, that the attractor pass through the Sun at 25.4 days and 4.57 billion years, and the track drawn for the slowest starter arrives at 26.6 days. What the figure cannot show is the saturated branch's physical cause: above a few days' rotation the dynamo stops responding to faster rotation, and that plateau is measured rather than derived.
Fig. 2 The torque the layer has been transmitting. Four stars braked from different starting rates converge onto a common track, and the Sun sits on it at 25 days having arrived from something like three. The convergence is what makes a rotation period an age, and it holds only if the whole star brakes together — a star whose interior kept its angular momentum would be braked at the surface and spun back up from below, and the period–age relation would be far looser than it is observed to be.

How the profile is actually measured

The measurement is an inversion, and it is worth understanding what is inverted and what is assumed.

The interior is read from a comb of frequencies. The Sun oscillates in millions of acoustic modes, each labelled by a radial order, a degree and an azimuthal order. In a spherically symmetric, non-rotating star, modes with the same radial order and degree but different azimuthal order have identical frequencies. Rotation breaks that degeneracy: a mode travelling with the rotation and one travelling against it have different frequencies, split by an amount proportional to a weighted average of the rotation rate over the region the mode occupies.

δνnℓm=m∫0R ⁣ ⁣∫0πKnℓm(r,θ) Ω(r,θ) dθ dr.\delta\nu_{n\ell m} = m\int_0^R\!\!\int_0^\pi K_{n\ell m}(r,\theta)\,\Omega(r,\theta)\,\mathrm{d}\theta\,\mathrm{d}r.

Each mode gives one such weighted average. The weighting — the kernel — is different for every mode, because each mode turns back at its own depth and has its own latitudinal extent.

Where each mode turns back: ℓ = 0 through the centre, ℓ = 300 in the outer 2 per cent. Why a set of frequencies is a depth profile and a single frequency is not. An acoustic wave travelling into a star meets a rising sound speed and is refracted back; it turns where its horizontal phase speed matches the local sound speed, which happens at c(r)/r = 2πν/√(ℓ(ℓ+1)). The horizontal axis is the angular degree on a logarithmic scale and the vertical axis is the fractional radius of that turning point, drawn at 2000, 3090, 4000 microhertz. The ordering is the content. A radial mode, ℓ = 0, has no horizontal phase speed at all and passes straight through the centre. Degrees one and two turn deep in the core. By ℓ = 300 the mode is trapped in the outer 2 per cent and knows nothing about anything below. So a frequency measured to a part in ten thousand constrains an average of the interior weighted in a way the mode itself decides, and measuring thousands of modes of different degree gives thousands of differently weighted averages — which is a solvable inverse problem, and is how the base of the convection zone was located at 0.713 of the radius rather than assumed. The sound speed here is a polytrope's rather than a tabulated solar model's, so the curve is the right shape and the wrong star in its outer tenth, where the real Sun is convective and this one is not. What the picture cannot show is the frequency dependence at fixed degree, which is weaker but not negligible: a higher-frequency mode of the same degree turns slightly deeper, and the three curves separating toward the right is that effect.
Fig. 3 The property that makes the inversion possible. Every note turns back at its own depth: a mode of low degree passes close to the centre and a mode of high degree is refracted back near the surface. So a set of modes samples the interior with thousands of differently weighted averages of the same function, and a linear combination of them can be constructed whose combined kernel is sharply peaked at any chosen depth. The rotation rate reported at 0.693 R is such a combination, and its resolution is limited by how well those kernels can be made to cancel.

Two things follow. First, the reported profile is not a measurement at a point but a localised average, and the width of that average — the resolution — is itself a computed quantity. Claiming a four-per-cent-thick tachocline requires showing that the averaging kernels are narrower than four per cent there, which they are for the equatorial regions and are not near the poles. The polar rotation rate below the convection zone is genuinely uncertain.

Second, the inversion needs a structural model. The kernels are computed from an assumed sound speed profile, so a structural error propagates into the rotation. The two are not independent, and a solar model that disagrees with helioseismology about its composition also disagrees, at some level, about its kernels.

One line, and a rotation rate at each end. Rotational splitting of mixed dipole modes in a red giant, against the fraction ζ of each mode's inertia that sits in the helium core. Every point is one multiplet; the scatter is a 0.011-nanohertz measurement error and is seeded so the drawing is reproducible. Because a mixed mode is a gravity wave in the core and a pressure wave in the envelope at once, and ζ says in what proportion, the splitting is a straight line in ζ whose value at ζ = 1 is the core's rotation and at ζ = 0 the envelope's. Fitting that line to the drawn points — rather than drawing the line the points were made from — returns a core period of 10.0 days and an envelope period of 165 days, against the 10 and 165 they were built from. The contrast is 16.5, and that is the number that does not fit: the core of a red giant has contracted by a factor of ten and the envelope has expanded by a hundred, so angular momentum conservation alone predicts a contrast of many hundreds. Something is coupling the two, and no mechanism proposed so far transports enough. What the figure cannot show is where between the two the transport happens, because ζ is a weighting and not a depth.
Fig. 4 The same technique applied to a star whose core turns twice as fast as the previous drawing’s. The splitting of the mixed modes rises in proportion, because a mode’s splitting is the rotation rate weighted by where the mode has amplitude — and the mixed modes have amplitude in the core. The measurement is linear in the rotation and the sensitivity is entirely a property of the mode, which is why a red giant’s core rotation is measurable at all and the Sun’s is measurable only because the Sun has p-modes reaching that deep.

What confines it

The candidate mechanisms are all magnetic, and they differ in where the field comes from. A fossil field in the radiative interior. If the radiative zone contains a large-scale field left over from the Sun’s formation, it would enforce rigid rotation there — a field line threading two shells at different rates is twisted, and a twisted field carries a torque until the twisting stops. The same field would resist the shear from above, confining the tachocline to the depth over which the field can be pushed aside. The difficulty is that such a field must not leak into the convection zone, where it would produce a surface signature nobody sees; keeping it buried requires the circulation from above to hold it down, and whether that works is a fine balance.

The dynamo’s own field. The oscillating field generated by the dynamo penetrates a short distance below the convection zone, and the associated stresses could confine the layer. The difficulty is timing: an oscillating field penetrates only a skin depth, and the skin depth for an eleven-year oscillation is far thinner than the tachocline. Nothing, and the spreading has been slower than calculated. The spreading estimate assumes a particular structure for the circulation, and stratification below the convection zone is strong enough that the estimate is sensitive to how the circulation is closed. This is the least satisfying answer and it has not been ruled out.

One line, and a rotation rate at each end. Rotational splitting of mixed dipole modes in a red giant, against the fraction ζ of each mode's inertia that sits in the helium core. Every point is one multiplet; the scatter is a 0.011-nanohertz measurement error and is seeded so the drawing is reproducible. Because a mixed mode is a gravity wave in the core and a pressure wave in the envelope at once, and ζ says in what proportion, the splitting is a straight line in ζ whose value at ζ = 1 is the core's rotation and at ζ = 0 the envelope's. Fitting that line to the drawn points — rather than drawing the line the points were made from — returns a core period of 20.0 days and an envelope period of 165 days, against the 20 and 165 they were built from. The contrast is 8.3, and that is the number that does not fit: the core of a red giant has contracted by a factor of ten and the envelope has expanded by a hundred, so angular momentum conservation alone predicts a contrast of many hundreds. Something is coupling the two, and no mechanism proposed so far transports enough. What the figure cannot show is where between the two the transport happens, because ζ is a weighting and not a depth.
Fig. 5 The same star with forty modes rather than twenty-two. More modes means more independent weighted averages, and the inversion’s resolution improves with them — but not indefinitely, because the kernels overlap and a fortieth mode adds less information than a fourth did. The resolution of an inversion saturates, and the thickness of the Sun’s tachocline is quoted as an upper bound for exactly this reason.

The rotation that is not steady

The profile drawn above is a time average. The Sun’s rotation is not constant, and the departures are small, measurable and unexplained in their own right.

The clearest of them are the torsional oscillations: bands of slightly faster and slightly slower rotation, a few metres a second in amplitude against a surface speed of two kilometres a second, which migrate towards the equator over the course of the magnetic cycle. They were found at the surface by Doppler measurements in the 1980s and traced downwards by helioseismology afterwards, and they reach a substantial fraction of the way through the convection zone.

Their significance is that they are in phase with the sunspot butterfly. The bands of faster rotation lead the latitude at which spots appear, so they are not a consequence of the spots but a precursor to them — which makes them a probe of the dynamo’s own wave rather than of its output. Whether they are the dynamo wave itself, or the flow’s response to the magnetic stresses the wave produces, is not settled.

A second and more contested variation concerns the tachocline directly. Analyses of the splitting data reported a periodic change in the rotation rate at the base of the convection zone with a period of about 1.3 years, appearing during one activity maximum and weakening afterwards. If real, it is a genuine oscillation of the layer this essay is about, and its period would constrain the field strength there.

The claim has been argued about for two decades. The signal is at the edge of what the inversions resolve, its amplitude is comparable with the systematic differences between analysis methods, and it has not been consistently recovered in later cycles. A detection that appears in one cycle and not the next is either a real intermittent phenomenon or an artefact of a changing data set, and distinguishing those requires a longer record than exists.

The general shape of that difficulty is worth naming, because it recurs wherever an inversion is pushed to its resolution limit. An inversion returns a localised average, and the width of the averaging kernel is a choice: a narrower kernel gives better spatial resolution and amplifies the noise, and a wider one does the reverse. Two groups making different choices produce different answers to the same question, and neither is wrong.

Where the feature being argued about is comparable in size with the kernel, the disagreement between methods is therefore not a measure of anybody’s care. It is the resolution limit expressing itself, and the honest way to report such a result is as a family of answers indexed by the smoothing chosen — which is what the better analyses do and what makes them harder to compare against the ones that do not.

The same caution applies to the four-per-cent thickness this essay opened with. That number is an upper bound set by the resolution rather than a measured width: what the inversions establish is that the transition is no thicker than the kernels can distinguish, which is thin enough to be a problem and is not a measurement of how thin.

Where each mode turns back: ℓ = 0 through the centre, ℓ = 300 in the outer 2 per cent. Why a set of frequencies is a depth profile and a single frequency is not. An acoustic wave travelling into a star meets a rising sound speed and is refracted back; it turns where its horizontal phase speed matches the local sound speed, which happens at c(r)/r = 2πν/√(ℓ(ℓ+1)). The horizontal axis is the angular degree on a logarithmic scale and the vertical axis is the fractional radius of that turning point, drawn at 2000, 3090, 4000 microhertz. The ordering is the content. A radial mode, ℓ = 0, has no horizontal phase speed at all and passes straight through the centre. Degrees one and two turn deep in the core. By ℓ = 300 the mode is trapped in the outer 2 per cent and knows nothing about anything below. So a frequency measured to a part in ten thousand constrains an average of the interior weighted in a way the mode itself decides, and measuring thousands of modes of different degree gives thousands of differently weighted averages — which is a solvable inverse problem, and is how the base of the convection zone was located at 0.713 of the radius rather than assumed. The sound speed here is a polytrope's rather than a tabulated solar model's, so the curve is the right shape and the wrong star in its outer tenth, where the real Sun is convective and this one is not. What the picture cannot show is the frequency dependence at fixed degree, which is weaker but not negligible: a higher-frequency mode of the same degree turns slightly deeper, and the three curves separating toward the right is that effect.
Fig. 6 Where each of twenty-two modes turns back. The lowest-degree modes reach the centre and the highest are confined to the outer few per cent, and everything between is a set of overlapping shells. An inversion is a weighted sum of these, and the width of the narrowest combination that can be built from them is the resolution the previous section’s argument is bounded by.

A chemical clock on the same layer

There is an independent constraint on how much material crosses the boundary, and it comes from an element.

Lithium is destroyed by proton capture at about two and a half million kelvin, which in the Sun is a little below the base of the convection zone. Any lithium mixed down past that depth is gone permanently. The Sun’s photospheric lithium abundance is a factor of about 140 below the meteoritic value, so a great deal of the convection zone’s lithium has been destroyed — which means material has been carried from the convection zone down past the burning depth, slowly, over billions of years.

That puts a number on the mixing across the tachocline: enough to deplete lithium by two orders of magnitude in 4.5 billion years, and not enough to disturb the sharpness of the rotation transition. Any confinement mechanism has to reproduce both, and the combination is more restrictive than either alone.

One line, and a rotation rate at each end. Rotational splitting of mixed dipole modes in a red giant, against the fraction ζ of each mode's inertia that sits in the helium core. Every point is one multiplet; the scatter is a 0.011-nanohertz measurement error and is seeded so the drawing is reproducible. Because a mixed mode is a gravity wave in the core and a pressure wave in the envelope at once, and ζ says in what proportion, the splitting is a straight line in ζ whose value at ζ = 1 is the core's rotation and at ζ = 0 the envelope's. Fitting that line to the drawn points — rather than drawing the line the points were made from — returns a core period of 20.0 days and an envelope period of 165 days, against the 20 and 165 they were built from. The contrast is 8.3, and that is the number that does not fit: the core of a red giant has contracted by a factor of ten and the envelope has expanded by a hundred, so angular momentum conservation alone predicts a contrast of many hundreds. Something is coupling the two, and no mechanism proposed so far transports enough. What the figure cannot show is where between the two the transport happens, because ζ is a weighting and not a depth.
Fig. 7 The same technique applied to a star where the answer is very different. In a red giant the modes are mixed — part gravity wave in the contracted core, part pressure wave in the vast envelope — so the splitting of each mode is a weighted average of two rotation rates with a weighting that shifts from mode to mode. Fitting the line returns both. The contrast that comes out is about ten; conservation of angular momentum through the contraction alone would give hundreds. The Sun’s problem and the giant’s are the same problem at two stages: transport that is too efficient to explain.
14 radial orders of the Sun, at 135.1 μHz apart. The p-mode spectrum of the Sun — 1 solar mass in 1 solar radius — from the asymptotic relation with its second-order term, drawn as 14 radial orders of ℓ = 0, 1 and 2 under a Gaussian envelope centred on ν_max = 3,090 μHz. Two numbers are marked and they do very different work. The large separation, 135.1 μHz, is the spacing between consecutive ℓ = 0 modes and fixes the mean density. The small separation, 9.00 μHz, is 2.7 pixels on this axis — it fixes the age, and it is why the échelle diagram exists rather than being a convenience. The vertical axis is the measurement: each mode moves the surface by about 20.0 cm s⁻¹ at the peak, and brightens it by a few parts per million, which is why this was impossible before a decade-long velocity series. Each mode is one line: its true width is set by its lifetime and is far below a pixel here.
Fig. 8 The raw material, over forty modes rather than the standard set. The comb is nearly regular — successive overtones are separated by a large frequency spacing that is a measure of the mean density — and the departures from regularity are where the structure is. Everything in this essay is a second-order property of this picture: the rotation comes from splittings within each multiplet, and the multiplets are invisible at this scale.

The near-surface layer, which is a second unexplained thing

There is a second feature of the profile that gets less attention and is equally unexplained.

In the outermost five per cent of the radius the rotation rate falls outwards at all latitudes — a shear layer at the top of the convection zone as well as at the bottom. It is shallower and less dramatic than the tachocline, and it is robustly measured, because the modes that sample it are the high-degree ones of which there are the most.

Nothing requires it. A convection zone in which turbulent transport were efficient and isotropic would rotate uniformly on cylinders or not at all; the observed profile is neither. The near-surface shear layer is generally attributed to the fact that the outermost convective cells are small and short-lived compared with the rotation period, so they do not feel the rotation and cannot maintain the differential pattern the deeper cells produce. That is a plausible story, and it has resisted being turned into a calculation that gets the depth and the amplitude right at the same time.

Two limitations of the inversion are worth stating together, because they bound what the tachocline debate can settle.

The first is a symmetry. A rotational splitting is odd in the azimuthal order mm, and what it measures is the component of the rotation that is symmetric about the equator. Any antisymmetric part — a north pole turning faster than the south — contributes nothing to first order and is simply invisible. The Sun’s magnetic activity is measurably asymmetric between hemispheres, so an asymmetric rotation is not an idle possibility, and every profile in this essay is an average over a symmetry that has not been checked.

The second is that the same technique applied to other stars returns an answer nobody can explain. Red giants have mixed modes that reach the core, and their splittings say the cores of these stars rotate perhaps ten times faster than the envelope — and a hundred times slower than any model of angular-momentum transport predicts for a star that has contracted its core by two orders of magnitude. Whatever couples the Sun’s interior to its convection zone is therefore not a solar peculiarity but a general mechanism, operating in evolved stars far more strongly than the candidate processes allow. The tachocline problem and the red-giant core-rotation problem are the same missing transport seen at two stages, and the second is the sharper of the two because the discrepancy is larger.

The flow that carries the field around

Every account of the solar cycle that places the dynamo in the tachocline needs something to return the field from the equator to the poles, and the candidate is a large-scale circulation in the convection zone: material flowing polewards near the surface, sinking, and returning equatorwards at depth.

The surface half of that flow is measured directly and is not controversial. A poleward flow of ten to twenty metres a second is seen in Doppler measurements and in the drift of surface magnetic features, and it is what carries the following polarity of each spot pair to the pole and reverses the polar field at each maximum.

The return flow is the problem. It has to exist — mass conserves — and where it flows and how fast decides the cycle period in an entire class of dynamo models, because the field is carried by it and the period is the transit time. Helioseismic techniques exist that are sensitive to it: measuring the difference in travel time between waves propagating north and south through a region gives the flow along that path.

The measurements have not converged. Different groups analysing overlapping data have reported a single circulation cell in each hemisphere, two cells stacked in radius, and more complicated arrangements, with return speeds differing by factors of several. The disagreements trace to how the systematic effects of the surface magnetic field on the wave travel times are removed, and to the fact that the signal is a fraction of a second in a travel time of an hour.

That is an uncomfortable position for a quantity that sets the period of the cycle in the leading model. A model whose most important parameter is measured to a factor of several is a model that cannot be falsified by its own prediction, and the flux-transport dynamos reproduce the eleven-year period by adopting a return speed within the observational range rather than by predicting one.

What the two measurements say together

The Sun says that transport across the radiative zone is efficient enough to have carried the surface’s braking to the centre and left no gradient. Red giants say that transport is efficient enough to keep a contracting core within a factor of ten of its envelope when the naive answer is hundreds. Both are statements that angular momentum moves through a stably stratified, non-convecting region far more freely than any hydrodynamic mechanism allows.

The candidates are the same in both cases: an internal magnetic field, or internal gravity waves generated at the convective boundary that propagate inwards and deposit their momentum where they break. Magnetic transport is efficient but tends to be too efficient — it enforces rigid rotation along field lines, which for the giants predicts a contrast of one where ten is measured. Wave transport is more adjustable and correspondingly harder to falsify.

14 radial orders of the Sun, at 135.1 μHz apart. The p-mode spectrum of the Sun — 1 solar mass in 1 solar radius — from the asymptotic relation with its second-order term, drawn as 14 radial orders of ℓ = 0, 1 and 2 under a Gaussian envelope centred on ν_max = 3,090 μHz. Two numbers are marked and they do very different work. The large separation, 135.1 μHz, is the spacing between consecutive ℓ = 0 modes and fixes the mean density. The small separation, 9.00 μHz, is 2.7 pixels on this axis — it fixes the age, and it is why the échelle diagram exists rather than being a convenience. The vertical axis is the measurement: each mode moves the surface by about 20.0 cm s⁻¹ at the peak, and brightens it by a few parts per million, which is why this was impossible before a decade-long velocity series. Each mode is one line: its true width is set by its lifetime and is far below a pixel here.
Fig. 9 The raw material for all of it. A star’s oscillation spectrum is a nearly regular comb of overtones, and everything in this essay is extracted from departures from that regularity — the splitting of each peak into a multiplet by rotation, and the deviation of the spacing from uniformity by structure. A four-year, uninterrupted photometric record resolves splittings of a few nanohertz, which is what the whole subject waited for.

It is worth naming the one measurement that would settle the confinement question, since three candidates have stood unresolved for two decades. A fossil field in the radiative interior would have an orientation, and an orientation that is not aligned with the rotation axis would make the rotation profile itself asymmetric — the very antisymmetric component the splittings cannot see. So the mechanism that is easiest to believe is the one the standard analysis is structurally blind to. Recovering it needs the even-order splitting coefficients, which are contaminated by the star’s own asphericity and by the magnetic field’s direct effect on the mode frequencies, and separating those three has not been done. The problem is therefore not short of data; it is short of an observable that distinguishes the candidates, which is a different and more stubborn kind of shortage. A field of the strength required is also well below what a direct Zeeman measurement of the solar interior could ever reach, since there is no line formed there to split, so the only access is through the frequencies themselves, and the frequencies respond to a field through its pressure rather than through its direction.

One line, and a rotation rate at each end. Rotational splitting of mixed dipole modes in a red giant, against the fraction ζ of each mode's inertia that sits in the helium core. Every point is one multiplet; the scatter is a 0.011-nanohertz measurement error and is seeded so the drawing is reproducible. Because a mixed mode is a gravity wave in the core and a pressure wave in the envelope at once, and ζ says in what proportion, the splitting is a straight line in ζ whose value at ζ = 1 is the core's rotation and at ζ = 0 the envelope's. Fitting that line to the drawn points — rather than drawing the line the points were made from — returns a core period of 20.0 days and an envelope period of 100 days, against the 20 and 100 they were built from. The contrast is 5.0, and that is the number that does not fit: the core of a red giant has contracted by a factor of ten and the envelope has expanded by a hundred, so angular momentum conservation alone predicts a contrast of many hundreds. Something is coupling the two, and no mechanism proposed so far transports enough. What the figure cannot show is where between the two the transport happens, because ζ is a weighting and not a depth.
Fig. 10 The same star with its envelope turning in a hundred days rather than a hundred and sixty-five, so the core-to-envelope contrast is smaller. The splittings of the p-dominated modes rise and the g-dominated ones do not move, which is the separation the whole method depends on: two families of mode in one spectrum, weighted towards two different regions, giving two rotation rates from one measurement. Nothing about the Sun’s own inversion is different in principle; it has more modes and less contrast.

Where the ladder goes

The immediate rung is the dynamo the tachocline is supposed to host: whether the solar cycle is generated there, in the shear, or distributed through the convection zone. The observational evidence is mixed, and fully convective stars — which have no tachocline at all — nevertheless have strong fields and cycles, which is the cleanest argument that a tachocline is not necessary for a dynamo.

The wider thread runs into every star with a radiative interior. The transport that is unexplained here decides how much processed material is mixed out of a burning core, which changes a star’s lifetime and its surface composition; it decides what rotation rate a stellar core has when it collapses, which decides the birth spin of a neutron star; and it decides whether a rotation period is a reliable age, since a surface decoupled from its interior would be braked and then spun back up.

Where this goes next

The Sun's internal rotation: differential above 0.693 R, rigid below, and the 15 nHz between two modes. Angular velocity against fractional radius, in nanohertz, from the inversion of hundreds of thousands of measured rotational splittings. Two features had to be discovered rather than deduced. The convection zone rotates differentially in latitude — 473 nanohertz at the equator, 330 at sixty degrees — and that latitude dependence persists all the way down through it, in surfaces that are very nearly radial rather than the cylinders a rotating convective fluid was expected to produce. And below about 0.693 of the radius the latitude dependence stops: the radiative interior turns as a single rigid body at 430 nanohertz, which is a remarkable thing for a fluid with no strength to do, and requires something — most likely a weak internal magnetic field — to be enforcing it. The two regimes are joined by a shear layer a few per cent of the radius thick, the tachocline, and it is where the solar magnetic cycle is generally thought to be generated, because it is the only place in the Sun with the shear a dynamo of that strength needs. The horizontal marks are what individual modes would report: the rotation averaged over the cavity each one occupies, weighted by the time the wave spends at each radius. A mode of degree 100 is trapped near the surface and reports 473 nanohertz; one of degree 1 passes through the deep interior and reports 458. The 15-nanohertz difference between them is the measurement, and splittings are measured to about a nanohertz — which is why the profile can be resolved at all. The kernels used here have the right support and the right sense; a real inversion uses computed eigenfunctions, and its resolution below 0.2 R is poor for the reason the previous figure gives. A fluid that turns as one piece The Sun's surface turns faster at its equator than at its poles, and everyone expected the inside to do something similar. Below seven-tenths of the way down it does not — the radiative interior rotates as rigidly as a bell, and nothing in hydrodynamics makes a fluid do that. 4 cycles of wings, and the polarity reverses at every boundary. Sunspot latitude against date, one mark per spot group, over 4 cycles of 11 years. The pattern is the reason the plot is called a butterfly diagram, and both of its features are laws with names. Spörer's law is the downward slope: spots emerge near ±28° at the start of a cycle and near ±7° at the end, so each wing narrows towards the equator and never crosses it. Hale's law is what the two mark shapes say: the leading spot of a pair has one magnetic polarity in the north and the other in the south, and both reverse when a new cycle starts — so a diagram that repeats every 11 years in appearance repeats only every 22 in magnetism. The wings overlap: the first high-latitude spots of a cycle appear about 1.6 years before the last low-latitude spots of the one before, which is why counting spots gives a cycle length slightly different from measuring one between polarity reversals. The dot density in time is the sunspot number itself, drawn from the standard skewed fitting function — the rise to maximum takes about four years and the decline about seven, in every cycle ever recorded. A magnetic clock read off a butterfly Plot sunspot latitude against date and the marks form wings that open at thirty degrees and march to the equator. The polarities reverse between wings, so the magnetic period is twenty-two years and the famous eleven is an artefact of counting spots rather than fields. Which gradient is steeper, and where. The temperature gradient radiation would need in order to carry the whole luminosity, against the gradient a rising blob of gas follows, through an n = 3 polytrope with a Kramers opacity and an energy generation going as ρT^4. Wherever the first exceeds the second the layer is unstable and convects. This model puts the outer boundary at 89 per cent of the radius; helioseismology measures the base of the Sun's convection zone at 71.3 per cent, and the gap between the two numbers is the price of a polytrope with one opacity law rather than a solar model. The curve also rises above the adiabatic value inside 4 per cent of the radius, which is a convective core — it appears when the energy generation is concentrated enough, and it is the structure a star burning by the CNO cycle has. The part of a star that boils Energy leaves a star's interior either by radiation or by bulk motion, and which one happens is settled by comparing two temperature gradients. The comparison decides that the Sun's outer third churns and its core does not, and that a massive star does exactly the opposite.

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Angular momentum transportConvective envelopeDifferential rotationHelioseismologyInternal magnetic fieldInversionLithium depletionMeridional circulationRadiative interiorRotational splittingSolar dynamoSpin-downTachoclineViscous diffusion