Mapping the Conditions for Hydrodynamic Instability on Steady State Accretion Models of Protoplanetary Disks

Hydrodynamical instabilities in disks around young stars depend on the thermodynamic stratification of the disk and on the local rate of thermal relaxation. Here, we map the spatial extent of unstable regions for the Vertical Shear Instability (VSI), the Convective OverStability (COS), and the amplification of vortices via the Subcritical Baroclinic Instability (SBI). We use steady state accretion disk models, including stellar irradiation, accretion heating and radiative transfer. We determine the local radial and vertical stratification and thermal relaxation rate in the disk, in dependence of the stellar mass, disk mass and mass accretion rate. We find that passive regions of disks - i.e. the midplane temperature dominated by irradiation - are COS unstable about one pressure scale height above the midplane and VSI unstable at radii $> 10 \, \text{au}$. Vortex amplification via SBI should operate in most parts of active and passive disks. For active parts of disks (midplane temperature determined by accretion power) COS can become active down to the midplane. Same is true for the VSI because of the vertically adiabatic stratification of an internally heated disk. If hydro instabilities or other non-ideal MHD processes are able to create $\alpha$-stresses ($> 10^{-5}$) and released accretion energy leads to internal heating of the disk, hydrodynamical instabilities are likely to operate in significant parts of the planet forming zones in disks around young stars, driving gas accretion and flow structure formation. Thus hydro-instabilities are viable candidates to explain the rings and vortices observed with ALMA and VLT.Comment: 24 pages, 13 figures, Accepted for publication in Ap

The Role of the Cooling Prescription for Disk Fragmentation: Numerical Convergence & Critical Cooling Parameter in Self-Gravitating Disks

Protoplanetary disks fragment due to gravitational instability when there is enough mass for self-gravitation, described by the Toomre parameter, and when heat can be lost at a rate comparable to the local dynamical timescale, described by t_c=beta Omega^-1. Simulations of self-gravitating disks show that the cooling parameter has a rough critical value at beta_crit=3. When below beta_crit, gas overdensities will contract under their own gravity and fragment into bound objects while otherwise maintaining a steady state of gravitoturbulence. However, previous studies of the critical cooling parameter have found dependence on simulation resolution, indicating that the simulation of self-gravitating protoplanetary disks is not so straightforward. In particular, the simplicity of the cooling timescale t_c prevents fragments from being disrupted by pressure support as temperatures rise. We alter the cooling law so that the cooling timescale is dependent on local surface density fluctuations, a means of incorporating optical depth effects into the local cooling of an object. For lower resolution simulations, this results in a lower critical cooling parameter and a disk more stable to gravitational stresses suggesting the formation of large gas giants planets in large, cool disks is generally suppressed by more realistic cooling. At our highest resolution however, the model becomes unstable to fragmentation for cooling timescales up to beta = 10.Comment: 10 pages, 6 figures. Accepted for publication in Ap

The Global Baroclinic Instability in Accretion Disks. II: Local Linear Analysis

This paper contains a local linear stability analysis for accretion disks under the influence of a global radial entropy gradient beta = - d log T / d log r for constant surface density. Numerical simulations suggested the existence of an instability in two- and three-dimensional models of the solar nebula. The present paper tries to clarify, quantify, and explain such a global baroclinic instability for two-dimensional flat accretion disk models. As a result linear theory predicts a transient linear instability that will amplify perturbations only for a limited time or up to a certain finite amplification. This can be understood as a result of the growth time of the instability being longer than the shear time which destroys the modes which are able to grow. So only non-linear effects can lead to a relevant amplification. Nevertheless, a lower limit on the entropy gradient ~beta = 0.22 for the transient linear instability is derived, which can be tested in future non-linear simulations. This would help to explain the observed instability in numerical simulations as an ultimate result of the transient linear instability, i.e. the Global Baroclinic Instability.Comment: 35 pages, 11 figures; ApJ in pres

High-resolution simulations of planetesimal formation in turbulent protoplanetary discs

We present high-resolution computer simulations of dust dynamics and planetesimal formation in turbulence generated by the magnetorotational instability. We show that the turbulent viscosity associated with magnetorotational turbulence in a non-stratified shearing box increases when going from 256^3 to 512^3 grid points in the presence of a weak imposed magnetic field, yielding a turbulent viscosity of $\alpha\approx0.003$ at high resolution. Particles representing approximately meter-sized boulders concentrate in large-scale high-pressure regions in the simulation box. The appearance of zonal flows and particle concentration in pressure bumps is relatively similar at moderate (256^3) and high (512^3) resolution. In the moderate-resolution simulation we activate particle self-gravity at a time when there is little particle concentration, in contrast with previous simulations where particle self-gravity was activated during a concentration event. We observe that bound clumps form over the next ten orbits, with initial birth masses of a few times the dwarf planet Ceres. At high resolution we activate self-gravity during a particle concentration event, leading to a burst of planetesimal formation, with clump masses ranging from a significant fraction of to several times the mass of Ceres. We present a new domain decomposition algorithm for particle-mesh schemes. Particles are spread evenly among the processors and the local gas velocity field and assigned drag forces are exchanged between a domain-decomposed mesh and discrete blocks of particles. We obtain good load balancing on up to 4096 cores even in simulations where particles sediment to the mid-plane and concentrate in pressure bumps.Comment: Accepted for publication in Astronomy & Astrophysics, with some changes in response to referee repor

Planet migration in three-dimensional radiative discs

The migration of growing protoplanets depends on the thermodynamics of the ambient disc. Standard modelling, using locally isothermal discs, indicate in the low planet mass regime an inward (type-I) migration. Taking into account non-isothermal effects, recent studies have shown that the direction of the type-I migration can change from inward to outward. In this paper we extend previous two-dimensional studies, and investigate the planet-disc interaction in viscous, radiative discs using fully three-dimensional radiation hydrodynamical simulations of protoplanetary accretion discs with embedded planets, for a range of planetary masses. We use an explicit three-dimensional (3D) hydrodynamical code NIRVANA that includes full tensor viscosity. We have added implicit radiation transport in the flux-limited diffusion approximation, and to speed up the simulations significantly we have newly adapted and implemented the FARGO-algorithm in a 3D context. First, we present results of test simulations that demonstrate the accuracy of the newly implemented FARGO-method in 3D. For a planet mass of 20 M_earth we then show that the inclusion of radiative effects yields a torque reversal also in full 3D. For the same opacity law used the effect is even stronger in 3D than in the corresponding 2D simulations, due to a slightly thinner disc. Finally, we demonstrate the extent of the torque reversal by calculating a sequence of planet masses. Through full 3D simulations of embedded planets in viscous, radiative discs we confirm that the migration can be directed outwards up to planet masses of about 33 M_earth. Hence, the effect may help to resolve the problem of too rapid inward migration of planets during their type-I phase.Comment: 16 pages, Astronomy&Astrophysics, in pres

Pebble trapping backreaction does not destroy vortices

The formation of planets remains one of the most challenging problems of contemporary astrophysics. Starting with micron-sized dust grains, coagulation models predict growth up to centimeter (pebbles), but growth beyond this size is difficult because of fragmentation and drift. Ways to bypass this problem have focused on inhomogeneities in the flow, be that zonal flows, streaming instability, or vortices. Because vortices are in equilibrium between the Coriolis and the pressure force, the pressureless grains will orbit along a vortex streamline experiencing a drag force. This is a very effective mechanism to concentrate pebbles as also seen in numerical simulations and possibly in ALMA observations. Yet, a high pebble load is dangerous for the vortex, and we showed that in two-dimensional simulations the backreaction eventually leads to vortex disruption. We investigate whether the same happens in three dimensions. We perform 3D simulations with pebbles in a local box finding that, although the pebbles disturb the vortex around the midplane, the column does not get destroyed. This result is important because, based on the previous 2D result suggesting complete disruption, the vortex interpretation of ALMA observations has been called into question. We show instead that the vortex behaves like a Taylor column, and the pebbles as obstacles to the flow. Pebble accumulation in the center of the vortices proceeds to roughly the same concentration as in the control run without backreaction.Comment: AAS research note; 3 pages, 1 figur

The fragmentation criteria in local vertically stratified self-gravitating disk simulations

Massive circumstellar disks are prone to gravitational instabilities, which trigger the formation of spiral arms that can fragment into bound clumps under the right conditions. Two dimensional simulations of self-gravitating disks are useful starting points for studying fragmentation, allowing for high-resolution simulations of thin disks. However, convergence issues can arise in 2D from various sources. One of these sources is the 2D approximation of self-gravity, which exaggerates the effect of self-gravity on small scales when the potential is not smoothed to account for the assumed vertical extent of the disk. This effect is enhanced by increased resolution, resulting in fragmentation at longer cooling timescales $\beta$. If true, it suggests that the 3D simulations of disk fragmentation may not have the same convergence problem and could be used to examine the nature of fragmentation without smoothing self-gravity on scales similar to the disk scale height. To that end, we have carried out local 3D self-gravitating disk simulations with simple $\beta$ cooling with fixed background irradiation to determine if 3D is necessary to properly describe disk fragmentation. Above a resolution of $\sim 40$ grid cells per scale height, we find that our simulations converge with respect to the cooling timescale. This result converges in agreement with analytic expectations which place a fragmentation boundary at $\beta_\mathrm{crit} = 3$.Comment: 11 pages, 9 figures. Accepted for publication in Ap