Linear and non-linear evolution of the vertical shear instability in accretion discs
01 Nov 2013-Monthly Notices of the Royal Astronomical Society (Oxford University Press)-Vol. 435, Iss: 3, pp 2610-2632
TL;DR: In this article, the authors analyse the stability and non-linear dynamics of power-law accretion disc models and present an accompanying stability analysis of the problem, based on asymptotic methods, that they use to guide their interpretation of the simulation results.
Abstract: We analyse the stability and non-linear dynamics of power-law accretion disc models. These have mid-plane densities that follow radial power laws and have either temperature or entropy distributions that are strict power-law functions of cylindrical radius, R. We employ two different hydrodynamic codes to perform high-resolution 2D axisymmetric and 3D simulations that examine the long-term evolution of the disc models as a function of the power-law indices of the temperature or entropy, the disc scaleheight, the thermal relaxation time of the fluid and the disc viscosity. We present an accompanying stability analysis of the problem, based on asymptotic methods, that we use to guide our interpretation of the simulation results. We find that axisymmetric disc models whose temperature or entropy profiles cause the equilibrium angular velocity to vary with height are unstable to the growth of perturbations whose most obvious character is modes with horizontal and vertical wavenumbers that satisfy vertical bar k(R)/k(Z)vertical bar 1. Instability occurs only when the thermodynamic response of the fluid is isothermal, or the thermal evolution time is comparable to or shorter than the local dynamical time-scale. These discs appear to exhibit the Goldreich-Schubert-Fricke or 'vertical shear' linear instability. Closer inspection of the simulation results uncovers the growth of two distinct modes. The first are characterized by very short radial wavelength perturbations that grow rapidly at high latitudes in the disc, and descend down towards the mid-plane on longer time-scales. We refer to these as 'finger modes' because they display k(R)/k(Z) 1. The second appear at slightly later times in the main body of the disc, including near the mid-plane. These 'body modes' have somewhat longer radial wavelengths. Early on they manifest themselves as fundamental breathing modes, but quickly become corrugation modes as symmetry about the mid-plane is broken. The corrugation modes are a prominent feature of the non-linear saturated state, leading to strong vertical oscillation of the disc mid-plane. In a viscous disc with aspect ratio H/r = 0.05, instability is found to operate when the viscosity parameter alpha < 4 x 10(-4). In three dimensions the instability generates a quasi-turbulent flow, and the associated Reynolds stress produces a fluctuating effective viscosity coefficient whose mean value reaches alpha similar to 10(-3) by the end of the simulation. The evolution and saturation of the vertical shear instability in astrophysical disc models which include realistic treatments of the thermal physics has yet to be examined. Should it occur on either global or local scales, however, our results suggest that it will have significant consequences for their internal dynamics, transport properties and observational appearance.
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TL;DR: In this article, a review of all aspects of planet formation by pebble accretion, from dust growth over planetesimal formation to the accretion of protoplanets and fully grown planets with gaseous envelopes, is presented.
Abstract: The detection and characterization of large populations of pebbles in protoplanetary disks have motivated the study of pebble accretion as a driver of planetary growth. This review covers all aspects of planet formation by pebble accretion, from dust growth over planetesimal formation to the accretion of protoplanets and fully grown planets with gaseous envelopes. Pebbles are accreted at a very high rate—orders of magnitude higher than planetesimal accretion—and the rate decreases only slowly with distance from the central star. This allows planetary cores to start their growth in much more distant positions than their final orbits. The giant planets orbiting our Sun and other stars, including systems of wide-orbit exoplanets, can therefore be formed in complete consistency with planetary migration. We demonstrate how growth tracks of planetary mass versus semimajor axis can be obtained for all the major classes of planets by integrating a relatively simple set of governing equations.
329 citations
TL;DR: In this article, the authors derived simple fitting formulae that feature all structural characteristics of protoplanetary discs during the evolution of several Myr and used them to identify preferred planetesimal and planet formation regions as a function of the disc's metallicity, accretion rate, and lifetime.
Abstract: The formation of planets with gaseous envelopes takes place in protoplanetary accretion discs on time scales of several million years. Small dust particles stick to each other to form pebbles, pebbles concentrate in the turbulent flow to form planetesimals and planetary embryos and grow to planets, which undergo substantial radial migration. All these processes are influenced by the underlying structure of the protoplanetary disc, specifically the profiles of temperature, gas scale height, and density. The commonly used disc structure of the minimum mass solar nebula (MMSN) is a simple power law in all these quantities. However, protoplanetary disc models with both viscous and stellar heating show several bumps and dips in temperature, scale height, and density caused by transitions in opacity, which are missing in the MMSN model. These play an important role in the formation of planets, since they can act as sweet spots for forming planetesimals via the streaming instability and affect the direction and magnitude of type-I migration. We present 2D simulations of accretion discs that feature radiative cooling and viscous and stellar heating, and they are linked to the observed evolutionary stages of protoplanetary discs and their host stars. These models allow us to identify preferred planetesimal and planet formation regions in the protoplanetary disc as a function of the disc's metallicity, accretion rate, and lifetime. We derive simple fitting formulae that feature all structural characteristics of protoplanetary discs during the evolution of several Myr. These fits are straightforward for applying to modelling any growth stage of planets where detailed knowledge of the underlying disc structure is required. (Less)
286 citations
TL;DR: In this paper, N-body simulations with synthetic forces from an underlying evolving gaseous disc were used to model the formation and long-term dynamical evolution of super-Earth systems.
Abstract: 'Hot super-Earths' (or 'mini-Neptunes') between one and four times Earth's size with period shorter than 100 d orbit 30-50 per cent of Sun-like stars Their orbital configuration - measured as the period ratio distribution of adjacent planets in multiplanet systems - is a strong constraint for formation models Here, we use N-body simulations with synthetic forces from an underlying evolving gaseous disc to model the formation and long-term dynamical evolution of super-Earth systems While the gas disc is present, planetary embryos grow and migrate inward to form a resonant chain anchored at the inner edge of the disc These resonant chains are far more compact than the observed super-Earth systems Once the gas dissipates, resonant chains may become dynamically unstable They undergo a phase of giant impacts that spreads the systems out Disc turbulence has no measurable effect on the outcome Our simulations match observations if a small fraction of resonant chains remain stable, while most super- Earths undergo a late dynamical instability Our statistical analysis restricts the contribution of stable systems to less than 25 per cent Our results also suggest that the large fraction of observed single-planet systems does not necessarily imply any dichotomy in the architecture of planetary systems Finally, we use the low abundance of resonances in Kepler data to argue that, in reality, the survival of resonant chains happens likely only in ~5 per cent of the cases This leads to a mystery: in our simulations only 50-60 per cent of resonant chains became unstable, whereas at least 75 per cent (and probably 90-95 per cent) must be unstable to match observations
283 citations
TL;DR: In this article, the authors present global MHD simulations of PPDs that include Ohmic resistivity and ambipolar diffusion, where the time-dependent gas-phase electron and ion fractions are computed under FUV and X-ray ionization with a simplified recombination chemistry.
Abstract: Protoplanetary disks (PPDs) are believed to accrete onto their central T Tauri star because of magnetic stresses. Recently published shearing box simulations indicate that Ohmic resistivity, ambipolar diffusion (AD) and the Hall effect all play important roles in disk evolution. In the presence of a vertical magnetic field, the disk remains laminar between 1–5 AU, and a magnetocentrifugal disk wind forms that provides an important mechanism for removing angular momentum. Questions remain, however, about the establishment of a true physical wind solution in the shearing box simulations because of the symmetries inherent in the local approximation. We present global MHD simulations of PPDs that include Ohmic resistivity and AD, where the time-dependent gas-phase electron and ion fractions are computed under FUV and X-ray ionization with a simplified recombination chemistry. Our results show that the disk remains laminar, and that a physical wind solution arises naturally in global disk models. The wind is sufficiently efficient to explain the observed accretion rates. Furthermore, the ionization fraction at intermediate disk heights is large enough for magneto-rotational channel modes to grow and subsequently develop into belts of horizontal field. Depending on the ionization fraction, these can remain quasi-global, or break-up into discrete islands of coherent field polarity. The disk models we present here show a dramatic departure from our earlier models including Ohmic resistivity only. It will be important to examine how the Hall effect modifies the evolution, and to explore the influence this has on the observational appearance of such systems, and on planet formation and migration.
276 citations
TL;DR: In this paper, the authors derived stringent limits on the non-thermal motions in the upper layers of the outer disk such that any contribution to the line-widths from turbulence is 30AU) disk.
Abstract: Turbulence can transport angular momentum in protoplanetary disks and influence the growth and evolution of planets. With spatially and spectrally resolved molecular emission line measurements provided by (sub)millimeter interferometric observations, it is possible to directly measure non-thermal motions in the disk gas that can be attributed to this turbulence. We report a new constraint on the turbulence in the disk around HD 163296, a nearby young A star, determined from ALMA Science Verification observations of four CO emission lines (the CO(3-2), CO(2-1), 13CO(2-1), and C18O(2-1) transitions). The different optical depths for these lines permit probes of non-thermal line-widths at a range of physical conditions (temperature and density) and depths into the disk interior. We derive stringent limits on the non-thermal motions in the upper layers of the outer disk such that any contribution to the line-widths from turbulence is 30AU) disk than has been previously considered.
254 citations
Cites background from "Linear and non-linear evolution of ..."
...In regions of weak MRI, a vertical shear instability may develop in the presence of varying angular velocity with height resulting in Reynolds stresses that drive α ∼ 10−3 (Nelson et al. 2013)....
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References
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TL;DR: In this article, a linear analysis is presented of the instability, which is local and extremely powerful; the maximum growth rate which is of the order of the angular rotation velocity, is independent of the strength of the magnetic field.
Abstract: A broad class of astronomical accretion disks is presently shown to be dynamically unstable to axisymmetric disturbances in the presence of a weak magnetic field, an insight with consequently broad applicability to gaseous, differentially-rotating systems. In the first part of this work, a linear analysis is presented of the instability, which is local and extremely powerful; the maximum growth rate, which is of the order of the angular rotation velocity, is independent of the strength of the magnetic field. Fluid motions associated with the instability directly generate both poloidal and toroidal field components. In the second part of this investigation, the scaling relation between the instability's wavenumber and the Alfven velocity is demonstrated, and the independence of the maximum growth rate from magnetic field strength is confirmed.
4,265 citations
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...…convective instability (Cameron & Pine 1973; Lin & Papaloizou 1980; Ruden, Papaloizou & Lin 1988; Ryu & Goodman 1992), gravitational instability (Toomre 1964; Lin & Pringle 1987; Papaloizou & Savonije 1991), the global and subcritical baroclinic instabilities (SBI; Klahr & Bodenheimer 2003;…...
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01 Jan 1990TL;DR: In this paper, the theory of the internal structure of stars and their evolution in time is introduced and the basic physics of stellar interiors, methods for solving the underlying equations, and the most important results necessary for understanding the wide variety of stellar types and phenomena.
Abstract: This book introduces the theory of the internal structure of stars and their evolution in time. It presents the basic physics of stellar interiors, methods for solving the underlying equations, and the most important results necessary for understanding the wide variety of stellar types and phenomena. The evolution of stars is discussed from their birth through normal evolution to possibly spectacular final stages. Chapters on stellar oscillations and rotation are included.
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...According to the standard theory of frequency-integrated radiative transfer with Rosseland mean opacities (Kippenhahn & Weigert 1990), the thermal time-scale due to radiative diffusion is tr = 2(3Cp/4ac)(ρ2κR /T 3), where κ R is the Rosseland mean opacity and c, a and Cp are the speed of light,…...
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TL;DR: In this paper, an approach to numerical convection is presented that exclusively yields upstream-centered schemes, which start from a meshwise approximation of the initial-value distribution by simple basic functions, e.g., Legendre polynomials.
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