Instability In Protoplanetary Discs Research Articles

Nonideal Magnetohydrodynamic Instabilities in Protoplanetary Disks: Vertical Modes and Reflection Asymmetry

Magnetized disk winds and wind-driven accretion are an essential and intensively studied dispersion mechanism of protoplanetary disks (PPDs). However, the stability of these mechanisms has yet to be adequately examined. This paper employs semi-analytic linear perturbation theories under nonideal magnetohydrodynamics, focusing on disk models whose magnetic diffusivities vary by a few orders of magnitude from the disk midplane to its surface. Linear modes are distinguished by their symmetry with respect to the midplane. These modes have qualitatively different growth rates: symmetric modes almost always decay, while at least one antisymmetric mode always has a positive growth rate. This growth rate decreases faster than the Keplerian angular velocity with cylindrical radius R in the disk and scales steeper than R −5/2 in the fiducial disk model. The growth of antisymmetric modes breaks the reflection symmetry across the disk equatorial plane, and may occur even in the absence of the Hall effect. In the disk regions where fully developed antisymmetric modes occur, accretion flows appear only on one side of the disk, while disk winds occur only on the other. This may explain the asymmetry of some observed PPD outflows.

A self-consistent model for dust settling and the vertical shear instability in protoplanetary disks

Abstract The spatial distribution of dust particles in protoplanetary disks affects dust evolution and planetesimal formation processes. The vertical shear instability (VSI) is one of the candidate hydrodynamic mechanisms that can generate turbulence in the outer disk region and affect dust diffusion. Turbulence driven by the VSI has a predominant vertical motion that can prevent dust settling. On the other hand, the dust distribution controls the spatial distribution of the gas cooling rate, thereby affecting the strength of VSI-driven turbulence. Here, we present a semi-analytic model that determines the vertical dust distribution and the strength of VSI-driven turbulence in a self-consistent manner. The model uses an empirical formula for the vertical diffusion coefficient in VSI-driven turbulence obtained from our recent hydrodynamical simulations. The formula returns the vertical diffusion coefficient as a function of the vertical profile of the cooling rate, which is determined by the vertical dust distribution. We use this model to search for an equilibrium vertical dust profile where settling balances with turbulent diffusion for a given maximum grain size. We find that if the grains are sufficiently small, there exists a stable equilibrium dust distribution where VSI-driven turbulence is sustained at a level of αz ∼ 10−3, where αz is the dimensionless vertical diffusion coefficient. However, as the maximum grain size increases, the equilibrium solution vanishes because the VSI can no longer stop the settling of the grains. This runaway settling may explain highly settled dust rings found in the outer part of some protoplanetary disks.

WInDI: a Warp-Induced Dust Instability in protoplanetary discs

ABSTRACT We identify a new dust instability that occurs in warped discs. The instability is caused by the oscillatory gas motions induced by the warp in the bending wave regime. We first demonstrate the instability using a local 1D (vertical) toy model based on the warped shearing box coordinates and investigate the effects of the warp magnitude and dust Stokes number on the growth of the instability. We then run 3D smoothed particle hydrodynamics (SPH) simulations and show that the instability is manifested globally by producing unique dust structures that do not correspond to gas pressure maxima. The 1D and SPH analysis suggest that the instability grows on dynamical time-scales and hence is potentially significant for planet formation.

Two saturated states of the vertical shear instability in protoplanetary disks with vertically varying cooling times

Abstract Turbulence in protoplanetary disks plays an important role in dust evolution and planetesimal formation. The vertical shear instability (VSI) is one of the candidate hydrodynamic mechanisms that can generate turbulence in the outer disk regions. The VSI requires rapid gas cooling in addition to vertical shear. A linear stability analysis suggests that the VSI may not operate around the midplane where gas cooling is inefficient. In this study, we investigate the nonlinear outcome of the VSI in disks with a linearly VSI-stable midplane region. We perform two-dimensional global hydrodynamical simulations of an axisymmetric disk with vertically varying cooling times. The vertical cooling time profile determines the thicknesses of the linearly VSI-stable midplane layer and unstable layers above and below the midplane. We find that the thickness of the midplane stable layer determines the vertical structure of VSI-driven turbulence in the nonlinear saturated state. We identify two types of final saturated state: (i) T states, characterized by vertical turbulent motion penetrating into the VSI-stable midplane layer, and (ii) pT states, characterized by turbulent motion confined in the unstable layers. The pT states are realized when the midplane VSI-stable layer is thicker than two gas scale heights. We also find that the VSI-driven turbulence is largely suppressed at all heights when the VSI-unstable region lying above and below the midplane is thinner than two gas scale heights. We present empirical formulas that predict the strength of VSI-driven turbulence as a function of the thicknesses of the unstable and stable layers. These formulas will be useful for studying how VSI-driven turbulence and dust grains controlling the disk cooling efficiency evolve simultaneously.

Nonlinear Outcome of Coagulation Instability in Protoplanetary Disks. II. Dust-ring Formation Mediated by Backreaction and Fragmentation

In our previous work (Paper I), we demonstrated that coagulation instability results in dust concentration against depletion due to the radial drift and accelerates dust growth locally. In this work (Paper II), we perform numerical simulations of coagulation instability taking into account effects of backreaction to gas and collisional fragmentation of dust grains. We find that the slowdown of the dust drift due to backreaction regulates dust concentration in the nonlinear growth phase of coagulation instability. The dust-to-gas surface density ratio increases from 10−3 up to ∼10−2. Each resulting dust ring tends to have a mass of ≃0.5 M ⊕ − 1.5 M ⊕ in our disk model. In contrast to Paper I, the dust surface density profile shows a local plateau structure at each dust ring. In spite of the regulation at the nonlinear growth, the efficient dust concentration reduces their collision velocity. As a result, dust grains can grow beyond the fragmentation barrier, and the dimensionless stopping time reaches unity, as in Paper I. The necessary condition for the efficient dust growth is (1) weak turbulence of α < 1 × 10−3 and (2) a large critical velocity for dust fragmentation (>1 m s−1). The efficient dust concentration in outer regions will reduce the inward pebble flux and is expected to decelerate the planet formation via the pebble accretion. We also find that the resulting rings can be unstable to secular gravitational instability (GI). The subsequent secular GI promotes planetesimal formation. We thus expect that a combination of these instabilities is a promising mechanism for dust-ring and planetesimal formation.

Continuing to hide signatures of gravitational instability in protoplanetary discs with planets

ABSTRACT We carry out 3D smoothed particle hydrodynamics simulations to study the impact of planet–disc interactions on a gravitationally unstable protoplanetary disc. We find that the impact of a planet on the disc’s evolution can be described by three scenarios. If the planet is sufficiently massive, the spiral wakes generated by the planet dominate the evolution of the disc and gravitational instabilities are completely suppressed. If the planet’s mass is too small, then gravitational instabilities are unaffected. If the planet’s mass lies between these extremes, gravitational instabilities are weakened. We present mock Atacama Large Millimeter/submillimeter Array (ALMA) continuum observations showing that the observability of large-scale spiral structures is diminished or completely suppressed when the planet is massive enough to influence the disc’s evolution. Our results show that massive discs that would be expected to be gravitationally unstable can appear axisymmetric in the presence of a planet. Thus, the absence of observed large-scale spiral structures alone is not enough to place upper limits on the disc’s mass, which could have implications on observations of young Class I discs with rings and gaps.

The coexistence of the streaming instability and the vertical shear instability in protoplanetary disks

The streaming instability is a promising mechanism to induce the formation of planetesimals. Nonetheless, this process has been found in previous studies to require either a dust-to-gas surface density ratio or a dust size that is enhanced compared to observed values. Employing two-dimensional global simulations of protoplanetary disks, we show that the vertical shear instability and the streaming instability in concert can cause dust concentration that is sufficient for planetesimal formation for lower surface density ratios and smaller dust sizes than the streaming instability in isolation, and in particular under conditions that are consistent with observational constraints. This is because dust overdensities forming in pressure bumps induced by the vertical shear instability act as seeds for the streaming instability and are enhanced by it. While our two-dimensional model does not include self-gravity, we find that strong dust clumping and the formation (and dissolution) of gravitationally unstable overdensities can be robustly inferred from the evolution of the maximum or the mean dust-to-gas volume density ratio. The vertical shear instability puffs up the dust layer to an average mid-plane dust-to-gas density ratio that is significantly below unity. We therefore find that reaching a mid-plane density ratio of one is not necessary to trigger planetesimal formation via the streaming instability when it acts in unison with the vertical shear instability.

Nonlinear Outcome of Coagulation Instability in Protoplanetary Disks. I. First Numerical Study of Accelerated Dust Growth and Dust Concentration at Outer Radii

Our previous linear analysis presents a new instability driven by dust coagulation in protoplanetary disks. The coagulation instability has the potential to concentrate dust grains into rings and assist dust coagulation and planetesimal formation. In this series of papers, we perform numerical simulations and investigate the nonlinear outcome of coagulation instability. In this paper (Paper I), we first conduct local simulations to demonstrate the existence of coagulation instability. Linear growth observed in the simulations is in good agreement with the previous linear analysis. We next conduct radially global simulations to demonstrate that coagulation instability develops during the inside-out disk evolution owing to dust growth. To isolate the various effects on dust concentration and growth, we neglect the effects of back-reaction to a gas disk and dust fragmentation in Paper I. This simplified simulation shows that neither back-reaction nor fragmentation is a prerequisite for local dust concentration via the instability. In most runs with weak turbulence, dust concentration via coagulation instability overcomes dust depletion due to radial drift, leading to the formation of multiple dust rings. The nonlinear development of coagulation instability also accelerates dust growth, and the dimensionless stopping time τ s reaches unity even at outer radii (>10 au). Therefore, coagulation instability is one promising process to retain dust grains and to accelerate dust growth beyond the drift barrier.

Coagulation Instability in Protoplanetary Disks: A Novel Mechanism Connecting Collisional Growth and Hydrodynamical Clumping of Dust Particles

Abstract We present a new instability driven by a combination of coagulation and radial drift of dust particles. We refer to this instability as “coagulation instability” and regard it as a promising mechanism to concentrate dust particles and assist planetesimal formation in the very early stages of disk evolution. Because of dust-density dependence of collisional coagulation efficiency, dust particles efficiently (inefficiently) grow in a region of positive (negative) dust density perturbations, leading to a small radial variation of dust sizes and as a result radial velocity perturbations. The resultant velocity perturbations lead to dust concentration and amplify dust density perturbations. This positive feedback makes a disk unstable. The growth timescale of coagulation instability is a few tens of orbital periods even when dust-to-gas mass ratio is on the order of 10−3. In a protoplanetary disk, radial drift and coagulation of dust particles tend to result in dust depletion. The present instability locally concentrates dust particles even in such a dust-depleted region. The resulting concentration provides preferable sites for dust–gas instabilities to develop, which leads to further concentration. Dust diffusion and aerodynamical feedback tend to stabilize short-wavelength modes, but do not completely suppress the growth of coagulation instability. Therefore, coagulation instability is expected to play an important role in setting up the next stage for other instabilities, such as streaming instability or secular gravitational instability, to further develop toward planetesimal formation.

The Sandwich Mode for Vertical Shear Instability in Protoplanetary Disks

Abstract Turbulence has a profound impact on the evolution of gas and dust in protoplanetary disks (PPDs), from driving the collisions and the diffusion of dust grains, to the concentration of pebbles in giant vortices, thus, facilitating planetesimal formation. The vertical shear instability (VSI) is a hydrodynamic mechanism, operating in PPDs if the local rate of thermal relaxation is high enough. Previous studies of the VSI have, however, relied on the assumption of constant cooling rates, or neglected the finite coupling time between the gas particles and the dust grains. Here, we present the results of hydrodynamic simulations of PPDs with the PLUTO code that include a more realistic thermal relaxation prescription, which enables us to study the VSI in the optically thick and optically thin parts of the disk under consideration of the thermal dust-gas coupling. We show the VSI to cause turbulence even in the optically thick inner regions of PPDs in our two- and three-dimensional simulations. The collisional decoupling of dust and gas particles in the upper atmosphere and the correspondingly inefficient thermal relaxation rates lead to the damping of the VSI turbulence. Long-lived anticyclonic vortices form in our three-dimensional simulation. These structures emerge from the turbulence in the VSI-active layer, persist over hundreds of orbits and extend vertically over the whole extent of the turbulent region. We conclude that the VSI leads to turbulence and the formation of long-lived dust traps within ±3 pressure scale heights distance from the disk midplane.

Stratified and Vertically Shearing Streaming Instabilities in Protoplanetary Disks

Abstract Under the right conditions, the streaming instability between imperfectly coupled dust and gas is a powerful mechanism for planetesimal formation as it can concentrate dust grains to the point of gravitational collapse. In its simplest form, the streaming instability can be captured by analyzing the linear stability of unstratified disk models, which represent the midplane of protoplanetary disks. We extend such studies by carrying out vertically global linear stability analyses of dust layers in protoplanetary disks. We find that the dominant form of instability in stratified dust layers is the one driven by the vertical gradient in the rotation velocity of the dust−gas mixture, but also requires partial dust−gas coupling. These vertically shearing streaming instabilities grow on orbital timescales and occur on radial length scales ∼ 10−3 H g, where H g is the local pressure scale height. The classic streaming instability, associated with the relative radial drift between dust and gas, occurs on radial length scales ∼ 10−2 H g, but has much smaller growth rates than vertically shearing streaming instabilities. Including gas viscosity is strongly stabilizing and leads to vertically elongated disturbances. We briefly discuss the potential effects of vertically shearing streaming instabilities on planetesimal formation.

Hiding Signatures of Gravitational Instability in Protoplanetary Disks with Planets

Abstract We carry out three-dimensional smoothed particle hydrodynamics simulations to show that a migrating giant planet strongly suppresses the spiral structure in self-gravitating disks. We present mock Atacama Large Millimeter/submillimeter Array (ALMA) continuum observations that show that in the absence of a planet, spiral arms due to gravitational instability are easily observed. Whereas in the presence of a giant planet, the spiral structures are suppressed by the migrating planet resulting in a largely axisymmetric disk with a ring and gap structure. Our modeling of the gas kinematics shows that the planet’s presence could be inferred, for example, using optically thin 13C16O. Our results show that it is not necessary to limit the gas mass of disks by assuming high dust-to-gas mass ratios in order to explain a lack of spiral features that would otherwise be expected in high-mass disks.

The Axisymmetric Streaming Instability in Protoplanetary Disks

Abstract We reexamine the streaming instability in protoplanetary disks under the axisymmetric geometry. We identify the roles of all the relevant physical processes and clarify the clustering mechanism of dust, offering a new physical interpretation. Through back-reaction, the dust density fluctuations in combination with the mean relative drift velocity provide a source for the gas velocity, which in turn drives the velocity and divergence of the dust flow. In the limit where the dust-to-gas ratio , the action of the Coriolis force on the radial gradient of the azimuthal dust velocity generates the divergence of the dust. In the opposite limit , the vertical gradient of the vertical dust velocity makes the main contribution to the dust divergence. The different dust clustering mechanisms at and are referred to as Mode I and Mode II, respectively. In both cases, the dust divergence further enhances the dust density fluctuations, resulting in a positive feedback loop. In Mode I (or Mode II), the growth rate is contributed by the mean azimuthal (or radial) drag force in the gas equation of motion, while the mean radial (or azimuthal) drag tends to reduce it. The instability makes a transition from Mode I to Mode II when the coupling between the perturbed gas and dust velocities is stronger than the Coriolis force in the gas equations, which occurs at and for Stokes numbers St &lt; 1 and St &gt; 1, respectively.

Dust settling instability in protoplanetary discs

ABSTRACT The streaming instability (SI) has been extensively studied in the linear and non-linear regimes as a mechanism to concentrate solids and trigger planetesimal formation in the mid-plane of protoplanetary discs. A related dust settling instability (DSI) applies to particles while settling towards the mid-plane. The DSI has previously been studied in the linear regime, with predictions that it could trigger particle clumping away from the mid-plane. This work presents a range of linear calculations and non-linear simulations, performed with fargo3d, to assess conditions for DSI growth. We expand on previous linear analyses by including particle size distributions and performing a detailed study of the amount of background turbulence needed to stabilize the DSI. When including binned size distributions, the DSI often produces converged growth rates with fewer bins than the standard SI. With background turbulence, we find that the most favourable conditions for DSI growth are weak turbulence, characterized by α ≲ 10−6 with intermediate-sized grains that settle from one gas scale height. These conditions could arise during a sudden decrease in disc turbulence following an accretion outburst. Ignoring background turbulence, we performed a parameter survey of local 2D DSI simulations. Particle clumping was either weak or occurred slower than particles settle. Clumping was reduced by a factor of 2 in a comparison 3D simulation. Overall, our results strongly disfavour the hypothesis that the DSI significantly promotes planetesimal formation. Non-linear simulations of the DSI with different numerical methods could support or challenge these findings.

The coexistence of the streaming instability and the vertical shear instability in protoplanetary disks

The streaming instability is a leading candidate mechanism to explain the formation of planetesimals. However, the role of this instability in the driving of turbulence in protoplanetary disks, given its fundamental nature as a linear hydrodynamical instability, has so far not been investigated in detail. We study the turbulence that is induced by the streaming instability as well as its interaction with the vertical shear instability. For this purpose, we employ the FLASH Code to conduct two-dimensional axisymmetric global disk simulations spanning radii from 1 to 100 au, including the mutual drag between gas and dust as well as the radial and vertical stellar gravity. If the streaming instability and the vertical shear instability start their growth at the same time, we find the turbulence in the dust midplane layer to be primarily driven by the streaming instability. The streaming instability gives rise to vertical gas motions with a Mach number of up to ~10−2. The dust scale height is set in a self-regulatory manner to about 1% of the gas scale height. In contrast, if the vertical shear instability is allowed to saturate before the dust is introduced into our simulations, then it continues to be the main source of the turbulence in the dust layer. The vertical shear instability induces turbulence with a Mach number of ~10−1 and thus impedes dust sedimentation. Nonetheless, we find the vertical shear instability and the streaming instability in combination to lead to radial dust concentration in long-lived accumulations that are significantly denser than those formed by the streaming instability alone. Therefore, the vertical shear instability may promote planetesimal formation by creating weak overdensities that act as seeds for the streaming instability.

Dense Particle Clouds in Laboratory Experiments in Context of Drafting and Streaming Instability

Abstract The streaming instability, as an example of instabilities driven by particle feedback on a gas flow, has been proven to have a major role in controlling the formation of planetesimals. These instabilities in protoplanetary disks occur at the transition from being gas-dominated to being dust and ice particle dominated. Here, we present experiments to approach this situation in the laboratory for particles in the Knudsen flow regime. In these experiments, we observe a particle cloud trapped for about 30 s in a rotating system under Earth’s gravity. For average dust-to-gas ratios up to 0.08, particles behave like individual test particles. Their sedimentation speed is identical to that of a single free-falling particle, even in locally denser regions. However, for higher dust-to-gas ratios, the motion of particles becomes sensitive to clumping. Particles in locally denser regions now sediment faster. Their sedimentation speed then depends linearly on the overall dust-to-gas ratio. This clearly shows a transition from tracerlike behavior to collective behavior. Beyond these findings, these types of experiments can now be used as a gauge to test particle feedback models in astrophysical hydrocodes, which are currently used for numerical simulations of streaming instabilities.

Identifying Anticyclonic Vortex Features Produced by the Rossby Wave Instability in Protoplanetary Disks

Abstract Several nearby protoplanetary disks have been observed to display large-scale crescents in the (sub)millimeter dust continuum emission. One interpretation is that these structures correspond to anticyclonic vortices generated by the Rossby wave instability within the gaseous disk. Such vortices have local gas overdensities and are expected to concentrate dust particles with a Stokes number around unity. This process might catalyze the formation of planetesimals. Whereas recent observations showed that dust crescents are indeed regions where millimeter-size particles have abnormally high concentration relative to the gas and smaller grains, no observations have yet shown that the gas within the crescent region counterrotates with respect to the protoplanetary disk. Here we investigate the detectability of anticyclonic features through measurement of the line-of-sight component of the gas velocity obtained with ALMA. We carry out 2D hydrodynamic simulations and 3D radiative transfer calculations of a protoplanetary disk characterized by a vortex created by the tidal interaction with a massive planet. As a case study, the disk parameters are chosen to mimic the IRS 48 system, which has the most prominent crescent observed to date. We generate synthetic ALMA observations of both the dust continuum and 12CO emission around the frequency of 345 GHz. We find that the anticyclonic features of the vortex are weak but can be detected if both the source and the observational setup are properly chosen. We provide a recipe for maximizing the probability of detecting such vortex features and present an analysis procedure to infer their kinematic properties.

Non-linear development of secular gravitational instability in protoplanetary disks

Abstract We perform non-linear simulation of secular gravitational instability (GI) in protoplanetary disks, which has been proposed as a mechanism of planetesimal and multiple ring formation. Since the timescale of the growth of the secular GI is much longer than the Keplerian rotation period, we develop a new numerical scheme for a long-term calculation utilizing the concept of symplectic integration. With our new scheme, we first investigate the non-linear development of the secular GI in a disk without a pressure gradient in the initial state. We find that the surface density of dust increases by more than a factor of 100 while that of gas does not increase even by a factor of 2, which results in the formation of dust-dominated rings. A line mass of the dust ring tends to be very close to the critical line mass of a self-gravitating isothermal filament. Our results indicate that the non-linear growth of the secular GI provides a powerful mechanism to concentrate the dust. We also find that the dust ring formed via the non-linear growth of the secular GI migrates inward with a low velocity, which is driven by the self-gravity of the ring. We give a semi-analytical expression for the inward migration speed of the dusty ring.

Vortices and the saturation of the vertical shear instability in protoplanetary discs

If sufficiently irradiated by its central star, a protoplanetary disks falls into an equilibrium state exhibiting vertical shear. This state may be subject to a hydrodynamical instability, the `vertical shear instability' (VSI), whose breakdown into turbulence transports a moderate amount of angular momentum while also facilitating planet formation, possibly via the production of small-scale vortices. In this paper, we show that VSI modes (a) exhibit arbitrary spatial profiles and (b) remain nonlinear solutions to the incompressible local equations, no matter their amplitude. The modes are themselves subject to parasitic Kelvin-Helmholtz instability, though the disk rotation significantly impedes the parasites and permits the VSI to attain large amplitudes (fluid velocities < 10% the sound speed). This `delay' in saturation probably explains the prominence of the VSI linear modes in global simulations. More generally, the parasites may set the amplitude of VSI turbulence in strongly irradiated disks. They are also important in breaking the axisymmetry of the flow, via the unavoidable formation of vortices. The vortices, however, are not aligned with the orbital plane and thus express a pronounced $z$-dependence. We also briefly demonstrate that the vertical shear has little effect on the magnetorotational instability, whereas magnetic fields easily quench the VSI, a potential issue in the ionised surface regions of the disk and also at larger radii.

Electron Heating and Saturation of Self-regulating Magnetorotational Instability in Protoplanetary Disks

Abstract Magnetorotational instability (MRI) has the potential to generatevigorous turbulence in protoplanetary disks, although its turbulence strength and accretion stress remain debatable because of the uncertainty of MRI with a low ionization fraction. We focus on the heating of electrons by strong electric fields, which amplifies nonideal magnetohydrodynamic effects. The heated electrons frequently collide with and stick to dust grains, which in turn decreases the ionization fraction and is expected to weaken the turbulent motion driven by MRI. In order to quantitatively investigate the nonlinear evolution of MRI, including the electron heating, we perform magnetohydrodynamical simulation with the unstratified shearing box. We introduce a simple analytic resistivity model depending on the current density by mimicking the resistivity given by the calculation of ionization. Our simulation confirms that the electron heating suppresses magnetic turbulence when the electron heating occurs with low current density. We find a clear correlation between magnetic stress and current density, which means that the magnetic stress is proportional to the squared current density. When the turbulent motion is completely suppressed, laminar accretion flow is caused by an ordered magnetic field. We give an analytical description of the laminar stateusing a solution of linear perturbation equations with resistivity. We also propose a formula that successfully predicts the accretion stress in the presence of the electron heating.