Global Disc Models Research Articles

Convective Overstability in Radially Global Protoplanetary Disks. I. Pure Gas Dynamics

Protoplanetary disks are prone to several hydrodynamic instabilities. One candidate, convective overstability (COS), can drive radial semiconvection that may influence dust dynamics and planetesimal formation. However, the COS has primarily been studied in local models. This paper investigates the COS near the midplane of radially global disk models. We first conduct a global linear stability analysis, which shows that linear COS modes exist only radially inward of their Lindblad resonance (LR). The fastest-growing modes have LRs near the inner radial domain boundary with effective radial wavelengths that can be a substantial fraction of the disk radius. We then perform axisymmetric global simulations and find that the COS’s nonlinear saturation is similar to previous incompressible shearing box simulations. In particular, we observe the onset of persistent zonal and elevator flows for sufficiently steep radial entropy gradients. In full 3D, nonaxisymmetric global simulations, we find the COS produces large-scale, long-lived vortices, which induce the outward radial transport of angular momentum via the excitation of spiral density waves. The corresponding α-viscosity values of order 10−3 agree well with those found in previous 3D compressible shearing box simulations. However, in global disks, significant modifications to their radial structure are found, including the formation of pressure bumps. Interestingly, the COS typically generates an outward radial mass transport, i.e., decretion. We briefly discuss the possible implications of our results for planetesimal formation and for interpreting dust rings and asymmetries observed in protoplanetary disks.

A Particle Module for the PLUTO Code. III. Dust

Abstract Implementation of a new particle module describing the physics of dust grains coupled to a gas via drag forces is the subject of this work. The proposed particle–gas hybrid scheme has been designed to work in Cartesian as well as in cylindrical and spherical geometries. The numerical method relies on a Godunov-type second-order scheme for the fluid and an exponential midpoint rule for dust particles, which overcomes the stiffness introduced by the linear coupling term. Besides being time-reversible and globally second-order accurate in time, the exponential integrator provides energy errors that are always bounded, and it remains stable in the limit of arbitrarily small particle stopping times, yielding the correct asymptotic solution. Such properties make this method preferable to the more widely used semi-implicit or fully implicit schemes at a very modest increase in computational cost. Coupling between particles and grid quantities is achieved through particle deposition and field-weighting techniques borrowed from particle-in-cell simulation methods. In this respect, we derive new weight factors in curvilinear coordinates that are more accurate than traditional volume or area weighting. A comprehensive suite of numerical benchmarks is presented to assess the accuracy and robustness of the algorithm in Cartesian, cylindrical, and spherical coordinates. Particular attention is devoted to the streaming instability, which is analyzed in both local and global disk models. The module is part of the PLUTO code for astrophysical gas dynamics, and it is mainly intended for the numerical modeling of protoplanetary disks in which solid and gas interact via aerodynamic drag.

Transient growth of perturbations on scales beyond the accretion disc thickness

Turbulent state of spectrally stable shear flows may be developed and sustained according to the bypass scenario of transition. If it works in non-magnetised boundless and homogeneous quasi-Keplerian flow, transiently growing shearing vortices should supply turbulence with energy. Employing the large shearing box approximation, as well as a set of global disc models, we study the optimal growth of the shearing vortices in such a flow in the whole range of azimuthal length-scales, $\lambda_y$, as compared to the flow scale-height, $H$. It is shown that with the account of the viscosity the highest possible amplification of shearing vortices, $G_{max}$, attains maximum at $\lambda_y\lesssim H$ and declines towards both the large scales $\lambda_y\gg H$ and the small scales $\lambda_y\ll H$ in a good agreement with analytical estimations based on balanced solutions. We pay main attention to the large-scale vortices $\lambda_y\gg H$, which produce $G_{max}\propto (\Omega/\kappa)^4$, where $\Omega$ and $\kappa$ denote local rotational and epicyclic frequencies, respectively. It is demonstrated that the large-scale vortices acquire high density perturbation as they approach the instant of swing. At the same time, their growth is not affected by bulk viscosity. We check that $G_{max}$ obtained globally is comparable to its local counterpart and the shape and localisation of global optimal vortices can be explained in terms of the local approach. The obtained results allow us to suggest that the critical Reynolds number of subcritical transition to turbulence in quasi-Keplerian flow, as well as the corresponding turbulent effective azimuthal stress should substantially depend on shear rate.

ON VERTICALLY GLOBAL, HORIZONTALLY LOCAL MODELS FOR ASTROPHYSICAL DISKS

Disks with a barotropic equilibrium structure, for which the pressure is only a function of the density, rotate on cylinders in the presence of a gravitational potential, so that the angular frequency of such a disk is independent of height. Such disks with barotropic equilibria can be approximately modeled using the shearing box framework, representing a small disk volume with height-independent angular frequency. If the disk is in baroclinic equilibrium, the angular frequency does generally depend on height, and it is thus necessary to go beyond the standard shearing box approach. In this paper, we show that given a global disk model, it is possible to develop approximate models that are local in horizontal planes without an expansion in height with shearing-periodic boundary conditions. We refer to the resulting framework as the vertically global shearing box (VGSB). These models can be non-axisymmetric for globally barotropic equilibria but should be axisymmetric for globally baroclinic equilibria. We provide explicit equations for this VGSB which can be implemented in standard magnetohydrodynamic codes by generalizing the shearing-periodic boundary conditions to allow for a height-dependent angular frequency and shear rate. We also discuss the limitations that result from the radial approximations that are needed in order to impose height-dependent shearing periodic boundary conditions. We illustrate the potential of this framework by studying a vertical shear instability and examining the modes associated with the magnetorotational instability.

Local and global aspects of the linear MRI in accretion discs

We revisit the linear MRI in a cylindrical model of an accretion disk and uncover a number of attractive results overlooked in previous treatments. In particular, we elucidate the connection between local axisymmetric modes and global modes, and show that a local channel flow corresponds to the evanescent part of a global mode. In addition, we find that the global problem reproduces the local dispersion relation without approximation, a result that helps explain the success the local analysis enjoys in predicting global growth rates. MRI channel flows are nonlinear solutions to the governing equations in the local shearing box. However, only a small subset of MRI modes share the same property in global disk models, providing further evidence that the prominence of channels in local boxes is artificial. Finally, we verify our results via direct numerical simulations with the Godunov code RAMSES.

GLOBAL SIMULATIONS OF PROTOPLANETARY DISKS WITH OHMIC RESISTIVITY AND AMBIPOLAR DIFFUSION

Protoplanetary disks 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 and the Hall effect all play important roles in disk evolution. In the presence of a vertical magnetic field, the disk remains laminar between 1-5au, 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 protoplanetary disks 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. 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.

Constraining the vertical structure of the Milky Way rotation by microlensing in a finite-width global disk model

In this paper we model the vertical structure of mass distribution of the Milky Way galaxy in the framework of a finite-width global disk model. Assuming the Galactic rotation curve only, we test inside the solar orbit the predictions of the model for two measurable and unrelated to each other processes: the gravitational microlensing that allows to fix the disk width-scale by the best fit to measurements, and the vertical gradient of rotation modelled in the quasi-circular orbits approximation. The former is sensitive to the gravitating mass in compact objects and the latter is sensitive to all kinds of gravitating matter. The analysis points to a small width-scale of the considered disks and, at most, insignificant contribution of non-baryonic dark mater in the solar circle. The predicted high vertical gradient values in the rotation are consistent with the gradient measurements.

Global simulations of magnetorotational turbulence – II. Turbulent energetics

Magnetorotational turbulence draws its energy from gravity and ultimately releases it via dissipation. However, the quantitative details of this energy flow have not been assessed for global disk models. In this work we examine the energetics of a well-resolved, three-dimensional, global magnetohydrodynamic accretion disk simulation by evaluating statistically-averaged mean-field equations for magnetic, kinetic, and internal energy using simulation data. The results reveal that turbulent magnetic (kinetic) energy is primarily injected by the correlation between Maxwell (Reynolds) stresses and shear in the (almost Keplerian) mean flow, and removed by dissipation. This finding differs from previous work using local (shearing-box) models, which indicated that turbulent kinetic energy was primarily sourced from the magnetic energy reservoir. Lorentz forces provide the bridge between the magnetic and kinetic energy reservoirs, converting ~ 1/5 of the total turbulent magnetic power input into turbulent kinetic energy. The turbulent energies (both magnetic and kinetic) are mainly driven by terms associated with the turbulent fields, with only a minor influence from mean magnetic fields. The interaction between mean and turbulent fields is most evident in the induction equation, with the mean radial magnetic field being strongly influenced by the turbulent electromotive force (EMF). During the quasi-steady turbulent state roughly 2/3 of the Poynting flux travels into the corona, with the remainder transporting magnetic energy in the radial direction. In contrast to previous studies, the stress-related part of the Poynting flux is found to dominate, which may have important implications for "reflection" models of Seyfert galaxy coronae that typically invoke a picture of buoyant rising of magnetic flux tubes via advection.

Global simulations of magnetorotational turbulence – I. Convergence and the quasi-steady state

Magnetorotational turbulence provides a viable mechanism for angular momentum transport in accretion disks. We present global, three dimensional (3D), MHD accretion disk simulations that investigate the dependence of the turbulent stresses on resolution. Convergence in the time-and-volume-averaged stress-to-gas-pressure ratio, at a value of $\sim0.04$, is found for a model with radial, vertical, and azimuthal resolution of 12-51, 27, and 12.5 cells per scale-height (the simulation mesh is such that cells per scale-height varies in the radial direction). A control volume analysis is performed on the main body of the disk (|z|<2H) to examine the production and removal of magnetic energy. Maxwell stresses in combination with the mean disk rotation are mainly responsible for magnetic energy production, whereas turbulent dissipation (facilitated by numerical resistivity) predominantly removes magnetic energy from the disk. Re-casting the magnetic energy equation in terms of the power injected by Maxwell stresses on the boundaries of, and by Lorentz forces within, the control volume highlights the importance of the boundary conditions (of the control volume). The different convergence properties of shearing-box and global accretion disk simulations can be readily understood on the basis of choice of boundary conditions and the magnetic field configuration. Periodic boundary conditions restrict the establishment of large-scale gradients in the magnetic field, limiting the power that can be delivered to the disk by Lorentz forces and by stresses at the surfaces. The factor of three lower resolution required for convergence in turbulent stresses for our global disk models compared to stratified shearing-boxes is explained by this finding. (Abridged)