A field the turbulence will not let in

A disc threaded by a vertical magnetic field can accrete without any turbulence at all. The field lines, anchored in the rotating gas and leaning outward by more than thirty degrees, fling material off the disc’s faces, and each gram that leaves carries away the angular momentum of several grams that stay. The turbulence that any field at all sets off through the magnetorotational instability becomes a secondary process, or none: a wind pulls on the whole face of the disc, the turbulence only across its edge, and the wind wins as soon as the field’s pressure is a few ten-thousandths of the gas’s.

That leaves the question the wind picture cannot answer for itself. Where does the field come from, and does the disc keep it? A disc’s vertical flux is not made in the disc. It is inherited, threaded through the gas that fell in from the surrounding cloud, and from then on it is moved about by the disc’s own flow. The accretion flow carries gas inward and drags field lines with it. The disc’s resistivity lets field lines slip through the gas. Which of the two wins decides whether the flux piles up near the star, where a wind needs it, or leaks away to the outer disc and beyond — and the answer, for a disc that accretes through turbulence, is uncomfortable.

The same eddies, twice #

Turbulence in a disc does two things that are usually discussed separately. Its stresses transport angular momentum outward, which lets the gas spiral inward, and in a thin-disc model that is represented by a viscosity $\nu$. And its eddies stir the field lines threaded through the gas, tangling and reconnecting them, which lets the large-scale field diffuse through the fluid as if the gas had a resistivity $\eta$. Both are properties of the same motions. The ratio of the two, the magnetic Prandtl number $\mathrm{Pm} = \nu/\eta$, is therefore not a property of the gas’s microphysics but of how the turbulence moves momentum compared with how it moves flux.

Simulations of magnetorotational turbulence have measured it, by imposing a weak large-scale field gradient and watching it decay, and they agree that the turbulent Prandtl number is of order one — between about one and a few. That is what one would guess: an eddy that carries momentum a given distance carries the field lines in it about as far.

The consequence follows from a single balance. The inflow drags field lines inward at the gas’s radial speed, $v_R = 3\nu/2R$ in a steady disc. The lines resist being dragged only through their bending: a line pinned far out and carried inward at the midplane is bent, and the bend has a radial component at the disc’s surface, $B_{R,s}$. Diffusion straightens the bend at a rate set by $\eta$ divided by the disc’s thickness $H$. In a steady state the two balance, and the field’s inclination at the surface comes out as

$$\tan i = \frac{B_{R,s}}{B_z} = \frac{v_R H}{\eta} = \frac{3}{2}\,\mathrm{Pm}\,\frac{H}{R}.$$

That is the first figure. Lubow, Papaloizou and Pringle derived it in 1994, and its point is the factor $H/R$. The inflow speed is slow because the disc is thin — the turbulence moves angular momentum over distances of order the thickness, while the gas has to travel the whole radius — and the diffusion is fast for the same reason: it has only the thickness to straighten. A turbulent disc bends its field by an angle of order its aspect ratio, a few degrees, and the thirty degrees a wind needs would require the turbulence to move momentum ten times more readily than it moves flux.

A field held at the right angle slips out #

The figure answers one question — how far a disc bends a field it is left alone with. The question a wind poses is the reverse. Suppose the field outside the disc is already arranged at thirty degrees, held there by the field’s own configuration above the disc. What does the disc do to it?

The field lines then move relative to the gas at the diffusion speed, $(\eta/H)\tan i$, against the inflow, and the net drift is $v_R - (\eta/H)\tan i$. With the angle held at thirty degrees and the Prandtl number near one, the outward term is larger by a factor of order $R/H$: at an aspect ratio of a twentieth the field slides outward at seven times the speed the gas moves inward. The flux leaves on a timescale shorter than the disc’s own accretion time by the same factor.

So the disc cannot have it both ways. A field gently bent by the inflow stays, but it is too upright to launch anything. A field leaning enough to launch a wind leaks away through the turbulence that the field itself would make unnecessary. For a disc that accretes by turbulence, the wind configuration is not a stable end state reached by accumulation; it would have to be imposed, and the disc would lose it.

How much flux there is to lose #

This would matter less if discs were short of flux. They are not. The gas that forms a disc arrives from a cloud core threaded by a field that was dynamically important there, and it brings far more flux than the disc could ever hold at the field strengths a wind needs.

A cloud core is held partly by its field, and its mass-to-flux ratio — measured in the units in which a ratio of one means the field can just support the cloud against gravity — is observed to be between one and a few. A solar mass at twice the critical ratio carries about $10^{30}$ gauss square centimetres. A disc with a field strong enough for a wind to dominate, plasma β near $10^4$, holds a few times $10^{27}$ inside a hundred astronomical units; even a disc whose field is a hundredth of its gas pressure holds a few per cent of what the core brought. And the star at the centre, with a kilogauss field over its own surface, holds less than a ten-thousandth.

This is the magnetic flux problem of star formation, seen from the disc’s side. Flux freezing would bring the core’s field into the star at ten thousand times its observed strength, and somewhere between the core and the star almost all of it is shed. Some is shed during the collapse, where the gas is so weakly ionised that the field slips through the neutrals by ambipolar diffusion. The rest has to be shed or kept by the disc, and the disc’s choice decides how it accretes. If the outward drift is as strong as the turbulent balance says, a disc that begins with a strong field quickly loses most of it, settles to a weak one, and accretes by turbulence. If the flux is retained, the disc is wind-driven.

Most of it is gone before there is a disc #

The flux a disc has to dispose of is not the core’s whole endowment, because much of it never arrives. A cloud core is held up partly by its field, and it collapses only once enough of the neutral gas has drifted through the field lines that its own gravity wins. That drift — ambipolar diffusion, the slow slippage of neutral molecules past the ions that are tied to the field — leaves flux behind in the envelope as mass moves in, and it is fastest exactly where the gas is densest and least ionised.

Near the forming star the density is so high and the ionisation so low that the field decouples almost entirely, and a region of a few tens of astronomical units across is left nearly unmagnetised while the field piles up just outside it. That decoupling is what lets a disc form at all. A field frozen into the collapsing gas all the way in would wind up as the material spun faster, and the torque of the wound field would brake the rotation so efficiently that no disc larger than a few astronomical units could survive — the magnetic braking problem of star formation. Discs of tens to hundreds of astronomical units are observed around protostars a hundred thousand years old, so something removed the field from the gas that made them — the same flux, carried into three different sizes, cannot all have stayed frozen.

The disc therefore starts with a fraction of the core’s flux that the collapse decided, and how large that fraction is depends on the ionisation chemistry — in particular on how many very small grains, which soak up electrons and ions, survived the collapse. The figure above gives the ceiling the disc must come under; the collapse decides how far under it starts.

A skin that conducts #

The argument so far treats the disc as a single slab with one viscosity and one resistivity. Real discs are layered, and the layering changes the balance.

Field lines move with whatever part of the gas they are most firmly attached to, and they are most firmly attached where the gas conducts best. The right average of the flow velocity is therefore weighted by conductivity, not by mass, and the diffusion term is reduced by the same weight. Guilet and Ogilvie worked through the consequences in 2012 and 2013. A disc whose surface layers are well ionised and laminar — not turbulent, so not diffusive — and which carries part of its accretion through those layers, drags its field inward through the skin even while the body lets it out.

That is exactly the situation a wind-driven disc is in. The surface of a protoplanetary disc is ionised by the star’s X-rays and ultraviolet; the wind’s torque acts there; and the accretion flow it drives is concentrated in a thin, fast layer near the surface, while the dense midplane is nearly laminar — the reverse of the picture in which a disc has to throw angular momentum outward through its body. A disc that already accretes through a wind may keep its flux. A disc that accretes through turbulence cannot acquire it. The two states are self-consistent separately, and the history of a disc — what flux it was born with, and how quickly its early turbulence bled it away — may decide which one it ends up in.

The simulations that follow the whole vertical structure are where this is being tested, and they disagree. Some find flux carried inward through the surface and concentrated near the star; others find it carried outward, with the wind itself removing it; the answer depends on the ionisation chemistry and on boundary conditions far above the disc that no observation constrains.

What a disc gains by keeping its flux, and where it must stop #

A disc that does accumulate flux gains the wind’s torque.

The torques cross at a β of a few thousand, and a factor of ten in field strength beyond that is a factor of a hundred in β and a hundredfold dominance of the wind. A disc that drags in even a modest fraction of the flux its cloud brought crosses into the wind-dominated regime without difficulty. But accumulating flux has a limit, and the limit is the instability itself.

The instability’s fastest mode grows longer as the field strengthens, and when it no longer fits inside the disc’s thickness — at β of about twenty — the turbulence switches off. A disc that has gathered that much flux is magnetically dominated: the field is strong enough to resist being wound up, the gas accretes through the wind or in streams along the field, and near a black hole such a state is thought to stop the inflow altogether until the field is pushed aside. Between the turbulent disc that cannot hold its flux and the magnetically arrested one that holds too much lies the range where a wind does the work and the turbulence is weak, and protoplanetary discs, if they accrete as the observations suggest, are somewhere in it.

The same balance around a black hole #

The argument is not specific to young stars, and its other extreme has been photographed. An accretion disc around a black hole is fully ionised, so the dead zones and the Hall effect are absent, but the balance between inward drag and turbulent slippage is the same, and so is the conclusion that a thin turbulent disc cannot hold a large-scale field. Thick discs are a different matter. When the gas is hot and the disc is nearly as thick as it is wide, $H/R$ is of order one and the outward slip is no faster than the inward drag. Flux brought in from the surroundings then accumulates at the centre until its pressure pushes back on the inflow — the magnetically arrested state, in which the field near the black hole is as strong as the inflow can confine.

The ring imaged around the black hole in M87 at a wavelength of 1.3 millimetres is polarised, and the pattern of its polarisation — the direction the electric vector twists around the ring — is what the arrested state predicts and what weakly magnetised models mostly do not. The same field, anchored in the black hole’s spinning spacetime, is a natural engine for the jet M87 drives across five thousand light years. If that reading is right, the black hole’s disc is the case of the balance in which the field won outright: a hot, thick flow that could not let its flux go, held there until the flux was as strong as the gas could bear. The thin, cool, turbulent discs around young stars sit at the other end, where the slip wins, and only the layering of their ionisation gives them a way back.

A field measured in a meteorite #

The strongest constraint on how much field the Sun’s own disc had comes from an unexpected direction. Chondrules — millimetre-sized droplets of rock that melted and cooled in the solar nebula before being swept into asteroids — recorded the ambient magnetic field in their iron-bearing grains as they cooled through their Curie points. Some of them have kept that magnetisation for four and a half billion years in meteorites that were never heated again. Laboratory measurements of chondrules from the Semarkona meteorite give a field of about fifty microtesla — half a gauss — in the region of the nebula where they formed, a few astronomical units from the young Sun.

Half a gauss in a disc of the minimum-mass density at a few astronomical units is a plasma β of order a few thousand to ten thousand, depending on the model of the nebula. That is the range in which a wind would carry the observed accretion rate, and a disc with turbulence alone would struggle to hold such a field against the outward drift drawn above. The measurement is one region, at one time, from one kind of grain, and the conversion from a remanent magnetisation to a field depends on how the grains acquired it. But it is the only direct measurement of a protoplanetary disc’s field anywhere, and it says the Sun’s disc kept a substantial part of its flux.

The present discs are harder. The polarisation of their millimetre dust emission, which in interstellar clouds traces the field, turns out in discs to be dominated by the scattering of their own radiation off large grains, and Zeeman measurements of molecular lines have so far given only upper limits of about a milligauss at tens of astronomical units. The velocity structure of jets, measured across their axes, gives the angular momentum they carry, and the rotation of a few has been measured — at values that need launching from radii of a fraction of an astronomical unit to a few, a magnetocentrifugal signature where it can be resolved.

What the slab argument leaves out #

The balance drawn in the first two figures uses a single viscosity, a single resistivity and a single bend through the disc’s thickness. Real discs have vertical structure in all three, and the conductivity-weighted average of the fourth figure is the first correction rather than the last. The turbulent Prandtl number is measured in shearing-box simulations of small patches of disc and may not apply to the large-scale field of a whole disc. The flux budget uses a minimum-mass disc and a single mass-to-flux ratio for the core. And protoplanetary discs are not fully ionised: ohmic, Hall and ambipolar effects make the resistivity a tensor that depends on the field itself, and in the Hall-dominated regions the drift depends on whether the field is aligned or anti-aligned with the rotation axis — so two discs differing only in the sign of their field transport flux differently.

What survives those corrections is the structure: the outward slip goes as $R/H$ times the inward drag whenever the same turbulence does both, and only a separation between where the gas flows and where the field diffuses can change that.

Still open: whether a disc’s flux goes in, out, or stays #

How much of its inherited flux a protoplanetary disc keeps, and where in the disc the flux ends up, decides whether its planets form in a turbulent disc or a laminar one, how fast they migrate, and how long the disc lives — and it is not known for any observed disc. The measurements that would decide it are the field itself, which millimetre Zeeman observations are approaching, and the radial distribution of accretion and wind mass loss, which the kinematics of disc winds seen in molecular emission are beginning to map. The meteorites have already said that one disc, the Sun’s, was threaded by a field of the strength a wind needs, a few astronomical units out, four and a half billion years ago. Whether that flux was carried in or merely failed to escape is the part they cannot say. Where it could not be kept, the disc stores its inflow instead.