On the convective overstability in protoplanetary discs

This paper explores the driving of low-level hydrodynamical activity in protoplanetary-disc dead zones. A small adverse radial entropy gradient, ordinarily stabilized by rotation, excites oscillatory convection (‘convective overstability’) when thermal diffusion, or cooling, is neither too strong nor too weak. I revisit the linear theory of the instability, discuss its prevalence in protoplanetary discs, and show that unstable modes are exact non-linear solutions in the local Boussinesq limit. Overstable modes cannot grow indefinitely, however, as they are subject to a secondary parametric instability that limits their amplitudes to relatively low levels. If parasites set the saturation level of the ensuing turbulence then the convective overstability is probably too weak to drive significant angular momentum transport or to generate vortices. But I also discuss an alternative, and far more vigorous, saturation route that generates radial ‘layers’ or ‘zonal flows’ (witnessed in semiconvection). Numerical simulations are required to determine which outcome is favoured in realistic discs, and consequently how important the instability is for disc dynamics.This research is partially funded by STFC grant ST/L000636/1.This is the final version of the article. It first appeared from Oxford University Press via http://dx.doi.org/10.1093/mnras/stv244

On dust–gas gravitational instabilities in protoplanetary discs

In protoplanetary discs the aerodynamical friction between particles and gas induces a variety of instabilities that facilitate planet formation. Of these we examine the so-called ‘secular gravitational instability’ (SGI) in the two-fluid approximation, deriving analytical expressions for its stability criteria and growth rates. Concurrently, we present a physical explanation of the instability that shows how it manifests upon an intermediate range of lengthscales exhibiting geostrophic balance in the gas component. The two-fluid SGI is completely quenched within a critical disc radius, as large as 10 au and 30 au for centimetre- and millimetre-sized particles, respectively, although establishing robust estimates is hampered by uncertainties in the parameters (especially the strength of turbulence) and deficiencies in the razor-thin disc model we employ. It is unlikely, however, that the SGI is relevant for well-coupled dust. We conclude by applying these results to the question of planetesimal formation and the provenance of large-scale dust rings.HNL acknowledges partial funding from Science and Technology Facilities Council (Grant ID: ST/L000636/1), and RR from a Bridgewater summer internship and from Newnham college

Spiral density waves and vertical circulation in protoplanetary discs

Spiral density waves dominate several facets of accretion disc dynamics – planet-disc interactions and gravitational instability (GI) most prominently. Though they have been examined thoroughly in two-dimensional simulations, their vertical structures in the non-linear regime are somewhat unexplored. This neglect is unwarranted given that any strong vertical motions associated with these waves could profoundly impact dust dynamics, dust sedimentation, planet formation, and the emissivity of the disc surface. In this paper, we combine linear calculations and shearing box simulations in order to investigate the vertical structure of spiral waves for various polytropic stratifications and wave amplitudes. For sub-adiabatic profiles, we find that spiral waves develop a pair of counter-rotating poloidal rolls. Particularly strong in the non-linear regime, these vortical structures issue from the baroclinicity supported by the background vertical entropy gradient. They are also intimately connected to the disc's g modes which appear to interact non-linearly with the density waves. Furthermore, we demonstrate that the poloidal rolls are ubiquitous in gravitoturbulence, emerging in the vicinity of GI spiral wakes, and potentially transporting grains off the disc mid-plane. Other than hindering sedimentation and planet formation, this phenomena may bear on observations of the disc's scattered infrared luminosity. The vortical features could also impact on the turbulent dynamo operating in young protoplanetary discs subject to GI, or possibly even galactic discs

Hydrodynamic convection in accretion discs

The prevalence and consequences of convection perpendicular to the plane of accretion discs have been discussed for several decades. Recent simulations combining convection and the magnetorotational instability have given fresh impetus to the debate, as the interplay of the two processes can enhance angular momentum transport, at least in the optically thick outburst stage of dwarf novae. In this paper we seek to isolate and understand the most generic features of disc convection, and so undertake its study in purely hydrodynamical models. First, we investigate the linear phase of the instability, obtaining estimates of the growth rates both semi-analytically, using one-dimensional spectral computations, as well as analytically, using WKBJ methods. Next we perform three-dimensional, vertically stratified, shearing box simulations with the conservative, finite-volume code PLUTO, both with and without explicit diffusion coefficients. We find that hydrodynamic convection can, in general, drive outward angular momentum transport, a result that we confirm with ATHENA, an alternative finite-volume code. Moreover, we establish that the sign of the angular momentum flux is sensitive to the diffusivity of the numerical scheme. Finally, in sustained convection, whereby the system is continuously forced to an unstable state, we observe the formation of various coherent structures, including large- scale and oscillatory convective cells, zonal flows, and small vortices

The vertical shear instability in poorly ionized, magnetized protoplanetary discs

Protoplanetary discs should exhibit a weak vertical variation in their rotation profiles. Typically this `vertical shear' issues from a baroclinic effect driven by the central star's radiation field, but it might also arise during the launching of a magnetocentrifugal wind. As a consequence, protoplanetary discs are subject to a hydrodynamical instability, the `vertical shear instability' (VSI), whose breakdown into turbulence could transport a moderate amount of angular momentum and facilitate, or interfere with, the process of planet formation. Magnetic fields may suppress the VSI, however, either directly via magnetic tension or indirectly through magnetorotational turbulence. On the other hand, protoplanetary discs exhibit notoriously low ionisation fractions, and non-ideal effects, if sufficiently dominant, may come to the VSI's rescue. In this paper we develop a local linear theory that explores how non-ideal MHD influences the VSI, while also launching additional diffusive shear instabilities. We derive a set of analytical criteria that establish when the VSI prevails, and then show how it can be applied to a realistic global model of a protoplanetary disc. Our calculations suggest that within ~10au the VSI should have little trouble emerging in the main body of the disk, but beyond that, and in the upper regions of the disc, its onset depends sensitively on the size of the preponderant dust grains

Dissipative structures in magnetorotational turbulence

Via the process of accretion, magnetorotational turbulence removes energy from a disk's orbital motion and transforms it into heat. Turbulent heating is far from uniform and is usually concentrated in small regions of intense dissipation, characterised by abrupt magnetic reconnection and higher temperatures. These regions are of interest because they might generate non-thermal emission, in the form of flares and energetic particles, or thermally process solids in protoplanetary disks. Moreover, the nature of the dissipation bears on the fundamental dynamics of the magnetorotational instability (MRI) itself: local simulations indicate that the large-scale properties of the turbulence (e.g. saturation levels, the stress-pressure relationship) depend on the short dissipative scales. In this paper we undertake a numerical study of how the MRI dissipates and the small-scale dissipative structures it employs to do so. We use the Godunov code RAMSES and unstratified compressible shearing boxes. Our simulations reveal that dissipation is concentrated in ribbons of strong magnetic reconnection that are significantly elongated in azimuth, up to a scale height. Dissipative structures are hence meso-scale objects, and potentially provide a route by which large scales and small scales interact. We go on to show how these ribbons evolve over time --- forming, merging, breaking apart, and disappearing. Finally, we reveal important couplings between the large-scale density waves generated by the MRI and the small-scale structures, which may illuminate the stress-pressure relationship in MRI turbulence

The stress-pressure relationship in simulations of MRI-induced turbulence

We determine how MRI-turbulent stresses depend on gas pressure via a suite of unstratified shearing box simulations. Earlier numerical work reported only a very weak dependence at best, results that call into question the canonical alpha-disk model and the thermal stability results that follow from it. Our simulations, in contrast, exhibit a stronger relationship, and show that previous work was box-size limited: turbulent `eddies' were artificially restricted by the numerical domain rather than by the scale height. Zero-net-flux runs without physical diffusion coefficients yield a stress proportional to $P^{0.5}$, where P is pressure. The stresses are also proportional to the grid length and hence remain numerically unconverged. The same runs with physical diffusivities, however, give a result closer to an alpha-disk: the stress is proportional to $P^{0.9}$. Net-flux simulations without explicit diffusion exhibit stresses proportional to $P^{0.5}$, but stronger imposed fields weaken this correlation. In summary, compressibility is important for the saturation of the MRI, but the exact stress-pressure relationship is difficult to ascertain in local simulations because of numerical convergence issues and the influence of any imposed flux. As a consequence, the interpretation of thermal stability behaviour in local simulations is a problematic enterprise.Some of the simulations were run on the DiRAC Complexity system, operated by the University of Leicester IT Services, which forms part of the STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment is funded by BIS National E-Infrastructure capital grant ST/K000373/1 and STFC DiRAC Operations grant ST/K0003259/1. DiRAC is part of the UK National E-Infrastructure run. JR and HNL are partially funded by STFC grants ST/L000636/1 and ST/K501906/1. JG acknowledges support from the Max-Planck-Princeton Center for Plasma Physics.This is the final version of the article. It first appeared from Oxford University Press via http://dx.doi.org/10.1093/mnras/stv228

Local models of astrophysical discs

Local models of gaseous accretion discs have been successfully employed for decades to describe an assortment of small-scale phenomena, from instabilities and turbulence, to dust dynamics and planet formation. For the most part, they have been derived in a physically motivated but essentially ad hoc fashion, with some of the mathematical assumptions never made explicit nor checked for consistency. This approach is susceptible to error, and it is easy to derive local models that support spurious instabilities or fail to conserve key quantities. In this paper we present rigorous derivations, based on an asympototic ordering, and formulate a hierarchy of local models (incompressible, Boussinesq and compressible), making clear which is best suited for a particular flow or phenomenon, while spelling out explicitly the assumptions and approximations of each. We also discuss the merits of the anelastic approximation, emphasizing that anelastic systems struggle to conserve energy unless strong restrictions are imposed on the flow. The problems encountered by the anelastic approximation are exacerbated by the disc's differential rotation, but also attend non-rotating systems such as stellar interiors. We conclude with a defence of local models and their continued utility in astrophysical research.HNL is partly funded by STFC grant ST/L000636/1

MRI turbulence and thermal instability in accretion discs

A long-standing puzzle in the study of black hole accretion concerns the presence or not of thermal instability. Classical theory predicts that the encircling accretion disc is unstable, as do some self-consistent magnetohydrodynamic simulations of the flow. Yet observations of strongly accreting sources generally fail to exhibit cyclic or unstable dynamics on the expected time-scales. This paper checks whether turbulent fluctuations impede thermal instability. It also asks if it makes sense to conduct linear stability analyses on a turbulent background. These issues are explored with a set of MRI simulations in thermally unstable local boxes in combination with stochastic equations that approximate the disc energetics. These models show that the disc’s thermal behaviour deviates significantly from laminar theory, though ultimately a thermal runaway does occur. We find that the disc temperature evolves as a biased random walk, rather than increasing exponentially, and thus generates a broad spread of outcomes, with instability often delayed for several thermal times. We construct a statistical theory that describes some of this behaviour, emphasizing the importance of the ‘escape time’ and its associated probability distribution. In conclusion, turbulent fluctuations on their own cannot stabilize a disc, but they can weaken and delay thermal instability.This work was partially funded by STFC grants ST/L000636/1 and ST/K501906/1. Some of the simulations were run on the DiRAC Complexity system, operated by the University of Leicester IT Services, which forms part of the STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment is funded by BIS National E- Infrastructure capital grant ST/K000373/1 and STFC DiRAC Operations grant ST/K0003259/1. DiRAC is part of the UK National E-Infrastructure