The weakest field changes the answer

Accretion is the most efficient energy source in the universe and for thirty years nobody could say what made it work. The problem was never where the energy came from; it was that matter in orbit has too much angular momentum to fall, some of it has to be carried outward, and no mechanism was available to carry it.

Molecular viscosity is far too small — by fourteen orders of magnitude in a protoplanetary disc. Turbulence would do it, but a Keplerian disc is stable against every hydrodynamic disturbance anybody could construct, and a stable flow does not become turbulent. The subject proceeded for three decades by writing the unknown stress as a parameter and fitting it.

The resolution turned out to be a result published in 1959 about a laboratory experiment, applied to discs in 1991, and it has the peculiar property that the agent responsible can be made arbitrarily weak without weakening the effect.

The criterion that says a disc is safe #

Rayleigh’s criterion for a rotating fluid is a statement about angular momentum. Displace a ring of fluid outward while it keeps its own angular momentum. If the ring now has less angular momentum than its new surroundings, it has too little centrifugal support, and it falls back: stable. If it has more, it keeps going: unstable.

Written out, the condition for stability is that specific angular momentum increases outward. A Keplerian disc has specific angular momentum proportional to the square root of the radius, which increases outward comfortably, so a Keplerian disc is stable — by a wide margin, at every radius, with no marginal cases.

The margin matters, because it is what made the situation so uncomfortable. This is not a flow sitting near a threshold where some neglected effect might tip it over; it is a flow that is stable by a factor of order unity at every point, against every axisymmetric hydrodynamic perturbation, in a criterion that has been checked for a century. The searches for a hydrodynamic route to turbulence in discs were not lazy. They were thorough, and they kept returning the same answer, which is that a Keplerian flow does not do it. That conclusion is correct and it is also, in a disc, irrelevant, because it assumes the displaced ring keeps its own angular momentum. A magnetic field breaks the assumption.

What a field line does between two rings #

Take two fluid elements at slightly different radii and imagine a field line threading both. In a well-conducting fluid that line is a material object: it is attached to both elements and it cannot be cut.

The inner element orbits faster. It runs ahead, and the field line connecting them stretches azimuthally. A stretched field line is under tension, and the tension pulls backwards on the leading element and forwards on the trailing one.

That is a torque, and it goes the wrong way. The inner element is slowed, loses angular momentum, and therefore sinks to a smaller radius, where it orbits faster still. The outer element is sped up, gains angular momentum, and rises. The separation between them grows, the field line stretches more, and the torque increases. Nothing about that loop depends on the tension being large.

The mechanical analogy that makes it obvious is two masses connected by a weak spring, orbiting at slightly different radii. Without the spring each keeps its own angular momentum and Rayleigh’s argument applies. With it, however weak, the inner mass is retarded and the outer advanced, and the configuration runs away — the spring’s weakness slows the approach to the runaway but does not prevent it, because the separation grows exponentially and the spring’s force grows with the separation. That is the whole of the mechanism, and it works for any non-zero spring constant.

What stops the runaway is the other thing tension does. A field line resists being bent as well as being stretched, and short-wavelength bends cost more than long ones. So a mode that is too short is stabilised by its own tension, and the wavelength at which that happens is proportional to the field strength. That is the one place the field’s magnitude enters.

So the criterion changes. What has to increase outward for stability is no longer the angular momentum but the angular velocity — and angular velocity falls outward in every gravitationally bound disc there is. Every one of them is unstable, and the instability is not marginal.

Three quarters, exactly #

The dispersion relation for this instability is a quadratic in the square of the growth rate, and it can be solved in closed form. For a Keplerian rotation law the positive root has a maximum at a wavenumber where the product of wavenumber and Alfvén speed equals root fifteen over four times the orbital frequency, and the growth rate there is exactly three quarters of the orbital frequency.

Both of those numbers are exact and neither contains the field strength. The Alfvén speed appears only in the combination that fixes where the maximum sits, so halving the field halves the wavenumber and doubles the wavelength while leaving the growth rate untouched.

A growth rate of three quarters of the orbital frequency means an e-folding in about a fifth of an orbit. At the radius of Jupiter that is a few years; close to a black hole it is milliseconds. On any timescale that matters for accretion, this is instantaneous.

The independence from field strength deserves a second look, because it is the feature that makes the result useful rather than merely true. Almost every instability in astrophysics has a threshold: a convective one needs a superadiabatic gradient, a gravitational one needs a mass above a Jeans value, a thermal one needs a cooling function with the wrong slope. Thresholds require the environment to be measured before the instability can be predicted. This one has no threshold in the field, so it can be asserted of a disc nobody has measured — and asserted of the first discs in the universe, whose fields were whatever the primordial seed happened to be.

The discontinuity in the limit is what makes this hard to see by intuition and easy to see by algebra. Set the field to zero and the instability vanishes. Set it to anything else, however small, and the growth rate is three quarters. There is no smooth approach; a hydrodynamic disc and a barely magnetised one behave completely differently.

Where the argument was hiding for thirty years #

The result was not new in 1991. Evgeny Velikhov derived the instability in 1959 for a magnetised Couette flow between rotating cylinders, and Subrahmanyan Chandrasekhar included it in his 1961 book on hydrodynamic and hydromagnetic stability. It was known, published, and correct.

What it lacked was an application. The laboratory context is a stability question about an apparatus, and the answer there is a nuisance to be designed around. Nobody connected it to accretion discs, and the accretion literature meanwhile developed the alpha prescription — a stress written as a parameter times the pressure — and got on with the consequences, which were considerable and mostly right.

Steven Balbus and John Hawley made the connection and stated it in the form that mattered: the criterion for a magnetised disc is different from Rayleigh’s, every astrophysical disc violates it, and the field required is negligible. That is a short paper’s worth of content and it reorganised the subject.

There is a lesson in the delay that is worth naming. The result was not found because nobody was looking for a stability result; the search was for a source of turbulence, and stability analyses are where one goes to prove that turbulence is absent. The paper that solved the problem was filed under the opposite heading for thirty-two years.

The second half of the lesson is about which assumption was doing the damage. Rayleigh’s criterion is not wrong and was never wrong; what was wrong was the unexamined premise that a displaced fluid element carries its angular momentum with it. In an unmagnetised fluid that premise is exact, so it never had to be stated, and an assumption that never has to be stated is one nobody thinks to relax. The same shape of error recurs whenever a criterion derived under one set of conditions is applied under another and the conditions are not part of the statement.

What the turbulence actually transports #

Growth is not transport. An instability that grows and saturates into turbulence carries angular momentum only if the turbulent motions are correlated in the right way, and whether they are is a question simulations answer rather than algebra.

They are. The saturated state carries an outward angular momentum flux, split between a Maxwell stress — the tension in the tangled field itself — and a Reynolds stress from correlated velocity fluctuations, in a ratio of about four to one.

The resulting alpha is around a hundredth in the simplest simulations, rising toward a tenth in some configurations. That is small, and smallness is the right answer: an alpha of order one would drain a disc in a few orbits, and discs are observed to last for millions of years — which is also what the depletion times measured across whole galaxies require of the gas on much larger scales.

It is worth being blunt about the status of this number. The growth rate is exact, the criterion is exact, the saturation amplitude is not — it depends on the field geometry assumed, on the resolution, on whether the simulation includes the vertical structure, and on the magnetic Prandtl number, which in a real disc is nothing like the value a simulation can afford. The mechanism is settled; the coefficient is a research programme.

One class of simulation result is worth flagging because it caused a genuine scare. In boxes with no net vertical flux threading them, the saturated stress was found to fall steadily as the resolution improved, with no sign of converging — which would mean the transport was an artefact of numerical dissipation rather than a physical result. Boxes with a net flux converge, and real discs are believed to have a net flux, so the practical answer is that the astrophysically relevant case is the well-behaved one. But the episode is a reminder that a number produced by a simulation is a number about the simulation until it has been shown not to depend on the simulation’s own scales.

There is an observational handle on alpha, and it is indirect. Dwarf novae outburst on a timescale set by how fast the disc drains, and matching those light curves requires an alpha of around a tenth in the hot state and a hundredth in the cold one. That is a measurement of a transport coefficient made from the shape of a light curve, and it is one of the few places the theory can be checked against a number rather than against a plausibility. The same light curves are what a companion feeding a compact object is usually detected by in the first place.

The part of a disc where none of this happens #

The instability needs the gas to be coupled to the field, and coupling requires charges. A protoplanetary disc’s midplane at a few astronomical units is at a hundred kelvin or less, shielded from the star’s ultraviolet by everything above it and from cosmic rays by a hundred grams per square centimetre of material, and its ionisation fraction falls to something like one part in ten to the thirteenth.

At that ionisation the magnetic Reynolds number falls to order unity, and the field simply diffuses through the gas rather than being carried by it. So a protoplanetary disc has a dead zone: a region around the midplane, roughly between a tenth of an astronomical unit and several, where the instability is suppressed and the transport is not what it is elsewhere. The surface layers above it stay ionised by X-rays and remain active. Material therefore accumulates at the dead zone’s edges, which is a pressure maximum, and a pressure maximum is where solid particles stop drifting inward — which is one of the more promising answers to why planetesimals form where they do.

That is a satisfying reversal. An instability whose defining feature is that it does not care how strong the field is turns out to care enormously about how many ions there are, and the astrophysical consequences follow from the chemistry rather than from the magnetism.

The chemistry is genuinely doing the work, and it is delicate. Removing the small grains from a disc raises the ionisation by orders of magnitude, because grains are where free electrons go to be neutralised; so a disc that has begun to grow its dust into pebbles becomes more magnetically active than one that has not, and the dead zone shrinks as the planet formation it enables proceeds. That is a feedback with the right sign to be interesting and the wrong level of certainty to be relied on, and it is currently one of the busier corners of the subject.

The instability’s two governing numbers are the shear index and the field strength, and each is worth reading over a wider range.

Two places the same argument reappears #

The instability is not confined to discs, and two other appearances are worth recording because they show which part of the argument is essential.

In the Sun’s radiative interior the rotation is nearly rigid, so the angular velocity gradient is almost zero and the instability is marginal — but not obviously absent, and whether it operates is one of the candidate explanations for why the tachocline has not spread. In a differentially rotating stellar interior the same criterion applies verbatim, and the shear is the whole question. And in the discs around neutron stars and black holes the same instability operates in a regime where the gas is fully ionised, the coupling is perfect, and the transport is unimpeded. Those are also the systems in which the efficiency is fixed by the inner edge alone, so the transport decides the rate of arrival and the geometry decides what each gram is worth. Those are the systems whose variability is most directly a picture of the turbulence, and the broad-band noise in an X-ray binary’s light curve is generally read as the superposition of fluctuations generated at every radius, each on that radius’s own orbital timescale.

That reading makes a prediction which is checkable and has been checked: the highest frequencies in the noise should correspond to the innermost radii, so a change in the inner edge should cut the noise off at a different frequency. The break frequency in the power spectra of accreting black holes does move as the source changes state, in the direction the picture requires. It is not a precision test, but it is a test, and it is one of the few ways the turbulence itself — rather than its integrated effect — is observed at all.

What an observer can actually see of it #

None of this is directly observable. The turbulence lives at a fraction of a scale height in a disc that is unresolved in almost every system it matters in, and the field strengths involved are far below anything a spectral line reports. What is observable is the consequence: matter arriving, and light coming off it. That is an uncomfortable position for a theory to be in, and it is worth stating plainly rather than glossing. The temperature run, the total luminosity and the efficiency all follow from energy conservation and the inner boundary alone; they would be the same whatever carried the angular momentum. The transport mechanism shows up only in the timescales — how fast a disc drains, how fast it responds to a change in supply, how much it varies — and timescales are the hardest things to measure well. The best current evidence that the mechanism is right is not a measurement of turbulence but the sheer breadth of what the picture accounts for: dwarf nova recurrence times, the state transitions of X-ray binaries, the lifetimes of protoplanetary discs, and the location of the pressure maxima where solids collect. Each of those was a separate puzzle, and each dissolves into a statement about how efficiently a magnetised disc moves angular momentum.

And two readings of the two quantities an observer or a simulation would have to measure.

What is exact and what is not #

It is worth separating the tiers, because this subject is unusual in having a very sharp core and a very soft periphery.

Exact: the criterion, which is that angular velocity must increase outward for stability, and which no astrophysical disc satisfies. Exact: the maximum growth rate of three quarters of the orbital frequency, and its independence from the field strength. Exact: the wavenumber at which that maximum occurs, and the cut-off at root three, both of which follow from the same quadratic. Not exact: the saturated stress, and therefore alpha; the field geometry the saturated state settles into; the behaviour when non-ideal effects — ambipolar diffusion, the Hall term, ohmic resistivity — are all present at once, which is the situation in most of a protoplanetary disc. The Hall term is particularly awkward, because its sign depends on whether the field is aligned or anti-aligned with the rotation, which means two otherwise identical discs behave differently according to the direction of a field nobody can measure.

The dependence on ionisation also means the instability is a diagnostic in the other direction: a disc observed to be transporting briskly is a disc that must be coupled, and therefore ionised, and therefore either warm or irradiated. The chemistry of a shielded core and the transport of a disc turn out to be the same question about cosmic rays asked at two densities.

The honest summary is that the question “why do discs accrete” has been answered and the question “how fast” has not.

There is a way to see why the second question is so much harder than the first. The criterion is a linear result about infinitesimal perturbations, and linear results are exact. The saturated stress is a property of a fully developed turbulent state, and turbulence has no exact results anywhere in physics — the same difficulty that leaves pipe flow and atmospheric convection parameterised rather than derived. Accretion is not unusual in having a well-understood instability and a poorly understood saturation; it is unusual in that the saturation is the number everything downstream needs. That is a considerable improvement on a parameter, and it is a smaller improvement than the elegance of the underlying result suggests.