The dynamical evolution of protoplanetary disks is of key interest for building a comprehensive theory of planet formation and to explain the observational properties of these objects. Using the magnetohydrodynamics code Athena++, with an isothermal shearing box setup, we study the boundary between the active and dead zone, where the accretion rate changes and mass can accumulate. We quantify how the turbulence level is affected by the presence of a non uniform ohmic resistivity in the radial-x direction that leads to a region of inhibited turbulence (or dead zone). Comparing the turbulent activity to that of ideal simulations, the turbulence inhibited area shows density fluctuations and magnetic activity at its boundaries, driven by energy injection from the active (ideal) zone boundaries. We find magnetic dissipation to be significantly stronger in the ideal regions, and the turbulence penetration through the boundary of the dead zone is determined by the value of the resistivity itself, through the ohmic dissipation process, though the thickness of the transition does not play a significant role in changing the dissipation. We investigate the 1D spectra along the shearing direction: magnetic spectra appear flat at large scales both in ideal as well as resistive simulations, though a Kolmogorov scaling over more than one decade persists in the dead zone, suggesting the turbulent cascade is determined by the hydrodynamics of the system: MRI dynamo action is inhibited where sufficiently high resistivity is present.
We conducted global hydrodynamic simulations of protoplanetary disk evolution with an adaptive Shakura-Sunyaev α prescription to represent the layered disk structure, and starting with the collapse phase of the molecular cloud. With the canonical values of model parameters, self-consistent dead zones formed at the scale of a few au. The instabilities associated with the dead zone and corresponding outbursts, similar to FUor eruptions, were also observed in the simulations.
Turbulence driven by magnetorotational instability (MRI) crucially affects the evolution of solid bodies in protoplanetary disks. On the other hand, small dust particles stabilize MRI by capturing ionized gas particles needed for the coupling of the gas and magnetic fields. To provide an empirical basis for modeling the coevolution of dust and MRI, we perform three-dimensional, ohmic-resistive MHD simulations of a vertically stratified shearing box with an MRI-inactive "dead zone" of various sizes and with a net vertical magnetic flux of various strengths. We find that the vertical structure o…
A number of transition disks exhibit significant azimuthal asymmetries in thermal dust emission. One possible origin for these asymmetries is dust trapping in vortices formed at the edges of dead zones. We carry out high-resolution, two-dimensional hydrodynamic simulations of this scenario, including the effects of dust feedback. We find that, although feedback weakens the vortices and slows down the process of dust accumulation, the dust distribution in the disk can nonetheless remain asymmetric for many thousands of orbits. We show that even after $10^4$ orbits, or $2.5$ Myr when scaled to t…
Protoplanetary discs are poorly ionised due to their low temperatures and high column densities, and are therefore subject to three "non-ideal" magnetohydrodynamic effects: Ohmic dissipation, ambipolar diffusion, and the Hall effect. The existence of magnetically driven turbulence in these discs has been a central question since the discovery of the magnetorotational instability. Early models considered Ohmic diffusion only and led to a scenario of layered accretion, in which a magnetically "dead" zone in the disc midplane is embedded within magnetically "active" surface layers at distances ~1…
Aims and Methods. Accretion bursts triggered by the magnetorotational instability (MRI) in the innermost disk regions were studied for protoplanetary gas-dust disks formed from prestellar cores of various mass $M_{\rm core}$ and mass-to-magnetic flux ratio $λ$. Numerical magnetohydrodynamics simulations in the thin-disk limit were employed to study the long-term ($\sim 1.0$~Myr) evolution of protoplanetary disks with an adaptive turbulent $α$-parameter, which depends explicitly on the strength of the magnetic field and ionization fraction in the disk. The numerical models also feature the co-e…
Global numerical simulations of protoplanetary disk formation and evolution were conducted in thin-disk limit, where the model included magnetically layered disk structure, a self-consistent treatment for the infall from cloud core as well as the smallest possible inner computational boundary. We compared the evolution of a layered disk with a fully magnetically active disk. We also studied how the evolution depends on the parameters of the layered disk model - the MRI triggering temperature and active layer thickness - as well as the mass of the prestellar cloud core. With the canonical value…
We study how the magnetorotational instability (MRI) in protoplanetary disks is affected by the electric discharge caused by the electric field in the resistive MHD. We have performed three-dimensional shearing box simulations with various values of plasma beta and electrical breakdown models. We find the self-sustainment of the MRI in spite of the high resistivity. The instability gives rise to the large electric field that causes the electrical breakdown, and the breakdown maintains the high ionization degree required for the instability. The condition for this self-sustained MRI is set by t…
Turbulence and angular momentum transport in accretion disks remains a topic of debate. With the realization that dead zones are robust features of protoplanetary disks, the search for hydrodynamical sources of turbulence continues. A possible source is the baroclinic instability (BI), which has been shown to exist in unmagnetized non-barotropic disks. We present shearing box simulations of baroclinicly unstable, magnetized, 3D disks, in order to assess the interplay between the BI and other instabilities, namely the magneto-rotational instability (MRI) and the magneto-elliptical instability. …
A protoplanetary disk typically forms a dead zone near its midplane at the distance of a few au from the central protostar. Accretion through such a magnetically layered disk can be intrinsically unstable and has been associated with episodic outbursts in young stellar objects. We present the first investigation into the effects of low metallicity environment on the structure of the dead zone as well as the resulting outbursting behavior of the protoplanetary disk. We conducted global numerical hydrodynamic simulations of protoplanetary disk formation and evolution in the thin-disk limit. The …
In Zombie Vortex Instability (ZVI), perturbations excite critical layers in stratified, rotating shear flow (as in protoplanetary disks), causing them to generate vortex layers, which roll-up into anticyclonic zombie vortices and cyclonic vortex sheets. The process is self-sustaining as zombie vortices perturb new critical layers, spawning a next generation of zombie vortices. Here, we focus on two issues: the minimum threshold of perturbations that trigger self-sustaining vortex generation, and the properties of the late-time zombie turbulence on large and small scales. The critical parameter…
[abridged] We present a 1+1D global magnetically-driven disk accretion model that captures the essence of the MRI-driven accretion, without resorting to 3D global non-ideal MHD simulations. The gas dynamics is assumed to be solely controlled by the MRI and hydrodynamic instabilities. For given stellar and disk parameters, the Shakura-Sunyaev viscosity parameter $α$ is determined self-consistently under the framework of viscously-driven accretion from detailed considerations of the MRI with non-ideal MHD effects (Ohmic resistivity and ambipolar diffusion), accounting for disk heating by stellar…
Planet formation at the inner edge of the dead zone I. the interplay between accretion outbursts and dust growth
Ludwig-Maximilians-Universität München（慕尼黑大学）；Max Planck Institute for Astronomy（马克斯·普朗克天文学研究所）；Center for Computational Astrophysics, Flatiron Institute（平顿研究所计算天体物理中心）
Azimuthally asymmetric structures have been discovered in millimeter continuum emission from many protoplanetary disks. One hypothesis is that they are vortices produced by the Rossby wave instability, for example at edges of planet-opened gaps or deadzones. Confirming the vortex nature of these structures will have profound implications to planet formation. One way to test the hypothesis is to compare the observed morphology of vortex candidates in near-infrared scattered light with theoretical expectations. To this end, we synthesize the appearance of vortices in H-band polarized light by co…
Sub-mm observations of protoplanetary disks now approach the acuity needed to measure the turbulent broadening of molecular lines. These measurements constrain disk angular momentum transport, and furnish evidence of the turbulent environment within which planetesimal formation takes place. We use local magnetohydrodynamic (MHD) simulations of the magnetorotational instability (MRI) to predict the distribution of turbulent velocities in low mass protoplanetary disks, as a function of radius and height above the mid-plane. We model both ideal MHD disks, and disks in which Ohmic dissipation resu…
Beyond the $α$ model: scaling the wind-driven accretion rate in protoplanetary disks using systematic non-ideal magnetohydrodynamical simulations
Department of Earth and Planetary Sciences, Institute of Science Tokyo（东京科学地球与行星科学系）；Institute for Advanced Study and Department of Astronomy, Tsinghua University（清华大学高等研究院天文学系）
Using radiation hydrodynamics simulations in a local stratified shearing box with realistic equations of state and opacities, we explored the outcome of self-gravity at 50 AU in a protoplanetary disc irradiated by the central star. We found that gravito-turbulence is sustained for a finite range of the surface density, from $\sim 80$ to $\sim$ 250 gcm$^{-2}$. The disk is laminar below the range while fragments above it. In the range of gravito-turbulence, the Toomre parameter decreases monotonically from $\sim 1$ to $\sim 0.7$ as the surface density increases while an effective cooling time is…
Close-in super-Earths are the most abundant exoplanets known. It has been hypothesized that they form in the inner regions of protoplanetary discs, out of the dust that may accumulate at the boundary between the inner region susceptible to the magneto-rotational instability (MRI) and an MRI-dead zone further out. In Paper I we presented a model for the viscous inner disc which includes heating due to both irradiation and MRI-driven accretion; thermal and non-thermal ionization; dust opacities; and dust effects on ionization. Here we examine how the inner disc structure varies with stellar, dis…
We analytically derive the expressions for the structure of the inner region of protoplanetary disks based on the results from the recent hydrodynamical simulations. The inner part of a disk can be divided into four regions: dust-free region with gas temperature in the optically thin limit, optically thin dust halo, optically thick condensation front and the classical optically thick region in order from the inside. We derive the dust-to-gas mass ratio profile in the dust halo using the fact that partial dust condensation regulates the temperature to the dust evaporation temperature. Beyond th…
There is considerable interest in hydrodynamic instabilities in dead zones of protoplanetary disks as a mechanism for driving angular momentum transport and as a source of particle-trapping vortices to mix chondrules and incubate planetesimal formation. We present simulations with a pseudo-spectral anelastic code and with the compressible code Athena, showing that stably stratified flows in a shearing, rotating box are violently unstable and produce space-filling, sustained turbulence dominated by large vortices with Rossby numbers of order 0.2-0.3. This Zombie Vortex Instability (ZVI) is obse…
Recent observations by the Atacama Large Millimeter/submillimeter Array (ALMA) of disks around young stars revealed distinct asymmetries in the dust continuum emission. In this work we want to study axisymmetric and non-axisymmetric structures, evocated by the magneto-rotational instability in the outer regions of protoplanetary disks. We combine the results of state-of-the-art numerical simulations with post-processing radiative transfer (RT) to generate synthetic maps and predictions for ALMA. We performed non-ideal global 3D MHD stratified simulations of the dead-zone outer edge using the F…
Magnetorotational instability (MRI) has a potential to generate the vigorous turbulence in protoplanetary disks, although its turbulence strength and accretion stress remains debatable because of the uncertainty of MRI with 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 …
A magnetic disc wind is an important mechanism that may be responsible for driving accretion and structure formation in protoplanetary discs. Recent numerical simulations have shown that these winds can take either the traditional `hourglass' symmetry about the mid-plane, or a `slanted' symmetry dominated by a mid-plane toroidal field of a single sign. The formation of this slanted symmetry state has not previously been explained. We use radially local 1D vertical shearing box simulations to assess the importance of large-scale MRI channel modes in influencing the formation and morphologies of…
The poorly-ionized interior of the protoplanetary disk is the location where dust coagulation processes may be most efficient. However even here, planetesimal formation may be limited by the loss of solid material through radial drift, and by collisional fragmentation of the particles. Our aim is to investigate the possibility that solid particles are trapped at local pressure maxima in the dynamically evolving disk. We perform the first 3-D global non-ideal MHD calculations of the disk treating the turbulence driven by the magneto-rotational instability. The domain contains an inner MRI-activ…
Gravitational instability (GI) controls the dynamics of young massive protoplanetary discs. Apart from facilitating gas accretion on to the central protostar, it must also impact on the process of planet formation: directly through fragmentation, and indirectly through the turbulent concentration of small solids. To understand the latter process, it is essential to determine the dust dynamics in such a turbulent flow. For that purpose, we conduct a series of 3D shearing box simulations of coupled gas and dust, including the gas's self-gravity and scanning a range of Stokes numbers, from 0.001 …
We study wakes and gap opening by low mass planets in gaseous protoplanetary disks threaded by net vertical magnetic fields which drive magnetohydrodynamical (MHD) turbulence through the magnetorotational instabilty (MRI), using three dimensional simulations in the unstratified local shearing box approximation. The wakes, which are excited by the planets, are damped by shocks similar to the wake damping in inviscid hydrodynamic (HD) disks. Angular momentum deposition by shock damping opens gaps in both MHD turbulent disks and inviscid HD disks even for low mass planets, in contradiction to the…
We improve on our previous treatments of long-term evolution of protostellar disks by explicitly solving disk self-gravity in two dimensions. The current model is an extension of the one-dimensional layered accretion disk model of Bae et al. We find that gravitational instability (GI)-induced spiral density waves heat disks via compressional heating (i.e. $P\rm{d}V$ work), and can trigger accretion outbursts by activating the magnetorotational instability (MRI) in the magnetically inert disk dead-zone. The GI-induced spiral waves propagate well inside of gravitationally unstable region before …
The magnetorotational instability (MRI) drives vigorous turbulence in a region of protoplanetary disks where the ionization fraction is sufficiently high. It has recently been shown that the electric field induced by the MRI can heat up electrons and thereby affect the ionization balance in the gas. In particular, in a disk where abundant dust grains are present, the electron heating causes a reduction of the electron abundance, thereby preventing further growth of the MRI. By using the nonlinear Ohm's law that takes into account electron heating, we investigate where in protoplanetary disks t…
The mechanism behind angular momentum transport in protoplanetary disks, and whether this transport is turbulent in nature, is a fundamental issue in planet formation studies. Recent ALMA observations have suggested that turbulent velocities in the outer regions of these disks are less than ~5-10% of the sound speed, contradicting theoretical predictions of turbulence driven by the magnetorotational instability (MRI). These observations have generally been interpreted to be consistent with a large-scale laminar magnetic wind driving accretion. Here, we carry out local, shearing box simulations…
Ionization-recombination balance in dense interstellar and circumstellar environments is a key factor for a variety of important physical processes, such as chemical reactions, dust charging and coagulation, coupling of the gas with magnetic field and development of instabilities in protoplanetary disks. We determine a critical gas density above which the recombination of electrons and ions on the grain surface dominates over the gas-phase recombination. For this regime, we present a self-consistent analytical model which allows us to exactly calculate abundances of charged species in dusty ga…
Turbulence in the protoplanetary disks induces collisions between dust grains, and thus facilitates grain growth. We investigate the two fundamental assumptions of the turbulence in obtaining grain collisional velocities -- the kinetic energy spectrum and the turbulence autocorrelation time -- in the context of the turbulence generated by the magneto-rotational instability (MRI). We carry out numerical simulations of the MRI as well as driven turbulence, for a range of physical and numerical parameters. We find that the convergence of the turbulence $α$-parameter does not necessarily imply the…
Recent sub-millimeter observations show non-axisymmetric brightness distributions with a horseshoe-like morphology for more than a dozen transition disks. The most accepted explanation for the observed asymmetries is the accumulation of dust in large-scale vortices. Protoplanetary disks vortices can form by the excitation of Rossby-wave instability in the vicinity of a steep pressure gradient, which can develop at the edges of a giant planet carved gap or at the edges of an accretionally inactive zone. We studied the formation and evolution of vortices formed in these two distinct scenarios by…
By constructing a global model based on 3D local magnetohydrodynamical (MHD) simulations, we show that the disk wind driven by magnetorotational instability (MRI) plays a significant role in the dispersal of the gas component of proto-planetary disks. Because the mass loss time scale by the MRI-driven disk winds is proportional to the local Keplerian rotation period, a gas disk dynamically evaporates from the inner region with possibly creating a gradually expanding inner hole, while a sizable amount of the gas remains in the outer region. The disk wind is highly time-dependent with quasi-peri…
MHD turbulence plays a crucial role in the dust dynamics of protoplanetary discs. It affects planet formation, vertical settling and is one possible origin of the large scale axisymmetric structures, such as rings, recently imaged by ALMA and SPHERE. Among the variety of MHD processes, the magnetorotational instability (MRI) has raised particular interest since it provides a source of turbulence and potentially organizes the flow into radial structures. However, the weak ionization of discs prevents the MRI from being excited beyond 1 AU. The strong sedimentation of millimetre dust measured in…
Non-ideal magnetohydrodynamical effects play a crucial role in determining the mechanism and efficiency of angular momentum transport as well as the level of turbulence in protoplanetary disks (PPDs), which are key to understanding PPD evolution and planet formation. It was shown in our previous work that at 1 AU, the magnetorotational instability (MRI) is completely suppressed when both Ohmic resistivity and ambipolar diffusion (AD) are taken into account, resulting in a laminar flow with accretion driven by magnetocentrifugal wind. In this work, we study the radial dependence of the laminar …
The gas surface density profile of protoplanetary disks is one of the most fundamental physical properties to understand planet formation. However, it is challenging to determine the surface density profile observationally, because the H$_2$ emission cannot be observed in low-temperature regions. We analyzed the Atacama Large Millimeter/submillimeter Array (ALMA) archival data of the \co line toward the protoplanetary disk around TW Hya and discovered extremely broad line wings due to the pressure broadening. In conjunction with a previously reported optically thin CO isotopologue line, the pr…
We perform 3D stratified shearing-box MHD simulations on the gas dynamics of protoplanetary disks threaded by net vertical magnetic field Bz. All three non-ideal MHD effects, Ohmic resistivity, the Hall effect and ambipolar diffusion are included in a self-consistent manner based on equilibrium chemistry. We focus on regions toward outer disk radii, from 5-60AU, where Ohmic resistivity tends to become negligible, ambipolar diffusion dominates over an extended region across disk height, and the Hall effect largely controls the dynamics near the disk midplane. We find that around R=5AU, the syst…
In this paper the migration of a 10 Earth-mass planetary core is investigated at the outer boundary of the dead zone of a protoplanetary disc by means of 2D hydrodynamic simulations done with the graphics processor unit version of the FARGO code. In the dead zone, the effective viscosity is greatly reduced due to the disc self-shielding against stellar UV radiation, X-rays from the stellar magnetosphere and interstellar cosmic rays. As a consequence, mass accumulation occurs near the outer dead zone edge, which is assumed to trap planetary cores enhancing the efficiency of the core-accretion s…
The aim of the present paper is to investigate the spatial structure of a protoplanetary disc whose dynamics is governed by magnetorotational turbulence. We perform a series of local 3D chemo-radiative MHD simulations located at different radii of a disc which is twice as massive as the standard minimum mass solar nebula of Hayashi (1981). The ionisation state of the disc is calculated by including collisional ionisation, stellar X-rays, cosmic rays and the decay of radionuclides as ionisation sources, and by solving a simplified chemical network which includes the effect of the absorption of …
We carry out a series of local, shearing box simulations of the outer regions of protoplanetary disks, where ambipolar diffusion is important due to low ionization levels, to better characterize the nature of turbulence and angular momentum transport in these disks. These simulations are divided into two groups, one with far ultraviolet (FUV) ionization included, and one without FUV. In both cases, we explore a large range in diffusivity values. We find that in the simulations without FUV, the properties of the turbulence are similar to the unstratified simulations of Bai & Stone (2011); for a…
While accreting through a circumstellar disk, young stellar objects are observed to undergo sudden and powerful accretion events known as FUor or EXor outbursts. Although such episodic accretion is considered to be an integral part of the star formation process, the triggers and mechanisms behind them remain uncertain. We conducted global numerical hydrodynamics simulations of protoplanetary disk formation and evolution in the thin-disk limit, assuming both magnetically layered and fully magnetorotational instability (MRI)-active disk structure. In this paper, we characterize the nature of the…
We apply ionization balance and MHD calculations to investigate whether magnetic activity moderated by recombination on dust can account for the mass accretion rates and the mid-infrared spectra and variability of protostellar disks. The MHD calculations use the stratified shearing-box approach and include grain settling and the feedback from the changing dust abundance on the resistivity of the gas. The two-decade spread in accretion rates among T Tauri stars is too large to result solely from variety in the grain size and stellar X-ray luminosity, but can be produced by varying these togethe…
We use local numerical simulations to study a vertically stratified accretion disk with a resistive mid-plane that damps magnetohydrodynamic (MHD) turbulence. This is an idealized model for the dead zones that may be present at some radii in protoplanetary and dwarf novae disks. We vary the relative thickness of the dead and active zones to quantify how forced fluid motions in the dead zone change. We find that the residual Reynolds stress near the mid-plane decreases with increasing dead zone thickness, becoming negligible in cases where the active to dead mass ratio is less than a few percen…
The gas dynamics of protoplanetary disks (PPDs) is largely controlled by non-ideal magnetohydrodynamic (MHD) effects including Ohmic resistivity, the Hall effect and ambipolar diffusion. Among these the role of the Hall effect is the least explored and most poorly understood. We have included all three non-ideal MHD effects in a self-consistent manner to investigate the role of the Hall effect on PPD gas dynamics using local shearing-box simulations. In this first paper, we focus on the inner region of PPDs, where previous studies excluding the Hall effect have revealed that the inner disk up …
The possible occurrence of dead zones in protoplanetary disks subject to the magneto-rotational instability highlights the importance of disk ionization. We present a closed-form theory for the deep-down ionization by X-rays at depths below the disk surface dominated by far-ultraviolet radiation. Simple analytic solutions are given for the major ion classes, electrons, atomic ions, molecular ions and negatively charged grains. In addition to the formation of molecular ions by X-ray ionization of H2 and their destruction by dissociative recombination, several key processes that operate in this …
We perform local, vertically stratified shearing-box MHD simulations of protoplanetary disks (PPDs) at a fiducial radius of 1 AU that take into account the effects of both Ohmic resistivity and ambipolar diffusion (AD). The magnetic diffusion coefficients are evaluated self-consistently from a look-up table based on equilibrium chemistry. We first show that the inclusion of AD dramatically changes the conventional picture of layered accretion. Without net vertical magnetic field, the system evolves into a toroidal field dominated configuration with extremely weak turbulence in the far-UV ioniz…
Turbulence in protoplanetary disks affects planet formation in many ways. While small dust particles are mainly affected by the aerodynamical coupling with turbulent gas velocity fields, planetesimals and larger bodies are more affected by gravitational interaction with gas density fluctuations. For the latter process, a number of numerical simulations have been performed in recent years, but a fully parameter-independent understanding has not been yet established. In this study, we present simple scaling relations for the planetesimal stirring rate in turbulence driven by magnetorotational in…
Excitation of Rossby wave instability and development of a large-scale vortex at the outer dead zone edge of protoplanetary discs is one of the leading theories that explains horseshoe-like brightness distribution in transition discs. Formation of such vortices requires a relatively sharp viscosity transition. Detailed modelling, however, indicates that viscosity transitions at the outer edge of the dead zone is relatively smooth. In this study, we present 2D global, non-isothermal, gas-dust coupled hydrodynamic simulations to investigate the possibility of vortex excitation at smooth viscosit…
The HD 163296 disk shows ring and gap substructures with ALMA observations. In addition, this is the only disk where the rings and gaps are spatially resolved in millimeter-wave polarization measurements. In this paper, we conduct radiative transfer modeling that includes self-scattering polarization to constrain the grain size and its distribution. We found that the grain size and dust scale height are the key parameters for reproducing the radial and azimuthal distributions of the observed polarization signature. Radial variation is mainly determined by grain size. The polarization fraction …
We carry out a series of local, vertically stratified shearing box simulations of protoplanetary disks that include ambipolar diffusion and a net vertical magnetic field. The ambipolar diffusion profiles we employ correspond to 30AU and 100AU in a minimum mass solar nebula (MMSN) disk model, which consists of a far-UV-ionized surface layer and low-ionization disk interior. These simulations serve as a follow up to Simon et al. (2013), in which we found that without a net vertical field, the turbulent stresses that result from the magnetorotational instability (MRI) are too weak to account for …
Wave interference as the origin of the cyclic magnetorotational dynamo in accretion disks: insights from weakly nonlinear theory and local shearing box simulations
Institute for Advanced Study（高等研究院）；Princeton University（普林斯顿大学）；Perimeter Institute for Theoretical Physics（基础物理研究所）
The magnetorotational instability (MRI) is the most promising mechanism by which angular momentum is efficiently transported outwards in astrophysical discs. However, its application to protoplanetary discs remains problematic. These discs are so poorly ionised that they may not support magnetorotational turbulence in regions referred to as `dead zones'. It has recently been suggested that the Hall effect, a non-ideal magnetohydrodynamic (MHD) effect, could revive these dead zones by enhancing the magnetically active column density by an order of magnitude or more. We investigate this idea by …
Vertical shear instability in two-moment radiation-hydrodynamical simulations of irradiated protoplanetary disks II. Secondary instabilities and stability regions
