P adé: A Code for Protoplanetary Disk Turbulence Based on Padé Differencing

Proto-planetary discs, the birth environment of planets, are an example of a structure commonly found in astrophysics, accretion discs. Identifying the mechanism responsible for accretion is a long-standing problem, dating back several decades. The common picture is that accretion is a consequence of turbulence, with several instabilities proposed for its origin. While traditionally this field used to be a purely theoretical endeavour, the landscape is now changing thanks mainly to new observational facilities such as the ALMA radio interferometer. Thanks to large improvements in spatial and spectral resolution and sensitivity (which have enabled the study of disc substructure, kinematics and surveys of large disc populations), multiple techniques have been devised to observationally measure the amount of turbulence in discs. This review summarises these techniques, ranging from attempts at direct detection of turbulence from line broadening, to more indirect approaches that rely on properties of the dust or consider the evolution of global disc properties (such as masses, radii and accretion rates) for large samples, and what their findings are. Multiple lines of evidence suggest that discs are in fact not as turbulent as thought one decade ago. On the other hand, direct detection of turbulence in some discs and the finite radial extent of dust substructures and in some cases the finite vertical extent strongly indicate that turbulence must be present at some level in proto-planetary discs. It is still an open question whether this amount of turbulence is enough to power accretion or if this is instead driven by other mechanisms, such as MHD winds.

The solution of the initial relative orbit determination problem using angles-only measurements is important for orbital proximity operations, satellite inspection and servicing, and the identification of unknown space objects in similar orbits. In this paper, a preliminary relative orbit determination performance analysis is conducted utilizing the linearized relative orbital equations of motion in cylindrical coordinates. The relative orbital equations of motion in cylindrical coordinates are rigorously derived in several forms included the exact nonlinear two-body differential equations of motion, the linear-time-varying differential equations of motion for an elliptical orbit chief, and the linear-time-invariant differential equations of motion for a circular orbit chief. Using the nonlinear angles-only measurement equation in cylindrical coordinates, evidence of full-relative-state observability is found, contrary to the range observability problem exhibited in Cartesian coordinates. Based on these results, a geometric approach to assess initial relative orbit determination performance is formulated. To facilitate a better understanding of the problem, the focus is on the 2-dimensional initial orbit determination problem. The results clearly show the dependence of the relative orbit determination performance on the geometry of the relative motion and on the time-interval between observations. Analysis is conducted for leader-follower orbits and flyby orbits where the deputy passes directly above or below the chief.

Growth of calcium–aluminum-rich inclusions by coagulation and fragmentation in a turbulent protoplanetary disk: Observations and simulations

We have studied dust evolution in a quiescent or turbulent protoplanetary disk by numerically solving a coagulation equation for settling dust particles, using the minimum mass solar nebula model. As a result, if we assume an ideally quiescent disk, the dust particles settle toward the disk midplane to form a gravitationally unstable layer within 2 × 103 to 4 × 104 yr at 1-30 AU, which is in good agreement with an analytic calculation by Nakagawa et al., although they did not take the particle size distribution into account explicitly. In an opposite extreme case of a globally turbulent disk, on the other hand, the dust particles fluctuate owing to turbulent motion of the gas and most particles become large enough to move inward very rapidly within 70 to 3 × 104 yr at 1-30 AU, depending on the strength of the turbulence. Our result suggests that global turbulent motion should cease for planetesimal formation to be possible in protoplanetary disks.

The stability properties of magnetized discs rotating with angular velocity Ω = Ω(s, z), dependent on both the radial and the vertical coordinates s and z, are considered. Such a rotation law is adequate for many astrophysical discs (e.g., galactic and protoplanetary discs, as well as accretion discs in binaries). In general, the angular velocity depends on height, even in thin accretion discs. A linear stability analysis is performed in the Boussinesq approximation, and the dispersion relation is obtained for short-wavelength perturbations. Any dependence of Ω on z can destabilize the flow. This concerns primarily small-scale perturbations for which the stabilizing effect of buoyancy is strongly suppressed due to the energy exchange with the surrounding plasma. For a weak magnetic field, instability of discs is mainly associated with vertical shear, whilst for an intermediate magnetic field the magnetic shear instability, first considered by Chandrasekhar and Velikhov, is more efficient. This instability is caused by the radial shear which is typically much stronger than the vertical shear. Therefore the growth time for the magnetic shear instability is much shorter than for the vertical shear instability. A relatively strong magnetic field can suppress both these instabilities. The vertical shear instability could be the source of turbulence in protoplanetary discs, where the conductivity is low.

We use magnetohydrodynamic simulations to measure relative speeds of solids in a protoplanetary disc with turbulence generated by the magnetorotational instability. Relative velocities are calculated as functions of particle Stokes number St, which measures the aerodynamic coupling to the gas. When relative velocities V rel are calculated between two particles i and j such that St i » St j and St j « 1, the data matches the analytical model of Ormel & Cuzzi. However, if V rel corresponds to two particles with the same St, only the data for the more loosely coupled solids (i.e. those with large St) follow the model. The discrepancy at the low-St end can be attributed to: (i) the numerical disc model's coarse resolution, which is unable to probe smaller turbulent eddies and, therefore, the dominant contribution to the particle relative velocities is given by the interpolation of the gas velocity inside the grid cells; (ii) the sparse particle sampling, which prevents the measurement of relative velocities between two particles in the same place at the same time. The distribution of turbulence-induced relative speeds can have a wide spread of values, which may lead to particle shattering, subject to the turbulent gas velocity. Codes such as the one used in this work, in general, underestimate relative velocities in turbulence for particles with St ≈ 1 because they lack energy on short time-scales (relative to a Kolmogorov spectrum). In making comparisons with theory, it is important to use the exact numerical energy spectrum instead of assuming a Kolmogorov inertial range.

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 instability (MRI), taking into account the stabilization of MRI due to Ohmic resistivity. We begin with order-of-magnitude estimates of the turbulence-induced gravitational force acting on solid bodies and associated diffusion coefficients for their orbital elements. We then test the predicted scaling relations using the results of recent Ohmic-resistive MHD simulations by Gressel et al. We find that these relations successfully explain the simulation results if we properly fix order-of-unity uncertainties within the estimates. We also update the saturation predictor for the density fluctuation amplitude in MRI-driven turbulence originally proposed by Okuzumi & Hirose. Combination of the scaling relations and saturation predictor allows to know how the turbulent stirring rate of planetesimals depends on disk parameters such as the gas column density, distance from the central star, vertical resistivity distribution, and net vertical magnetic flux. In Paper II, we apply our recipe to planetesimal accretion to discuss its viability in turbulent disks.

We perform a systematic study of the dynamics of dust particles in protoplanetary disks with embedded planets using global 2-D and 3-D inviscid hydrodynamic simulations. Lagrangian particles have been implemented into magnetohydrodynamic code Athena with cylindrical coordinates. We find two distinct outcomes depending on the mass of the embedded planet. In the presence of a low mass planet ($8 M_{\oplus}$), two narrow gaps start to open in the gas on each side of the planet where the density waves shock. These shallow gaps can dramatically affect particle drift speed and cause significant, roughly axisymmetric dust depletion. On the other hand, a more massive planet ($>0.1 M_{J}$) carves out a deeper gap with sharp edges, which are unstable to the vortex formation. Particles with a wide range of sizes ($0.02<\Omega t_{s}<20$) are trapped and settle to the midplane in the vortex, with the strongest concentration for particles with $\Omega t_{s}\sim 1$. The dust concentration is highly elongated in the $\phi$ direction, and can be as wide as 4 disk scale heights in the radial direction. Dust surface density inside the vortex can be increased by more than a factor of 10$^2$ in a very non-axisymmetric fashion. For very big particles ($\Omega t_{s}\gg 1$) we find strong eccentricity excitation, in particular around the planet and in the vicinity of the mean motion resonances, facilitating gap opening there. Our results imply that in weakly turbulent protoplanetary disk regions (e.g. the "dead zone") dust particles with a very wide range of sizes can be trapped at gap edges and inside vortices induced by planets with $M_{p}<M_{J}$, potentially accelerating planetesimal and planet formation there, and giving rise to distinctive features that can be probed by ALMA and EVLA.

The outer protoplanetary discs (PPDs) can be subject to the magnetorotational instability (MRI) and the vertical shear instability (VSI). While both processes can drive turbulence in the disc, existing numerical simulations have studied them separately. In this paper, we conduct global 3D non-ideal magnetohydrodynamic (MHD) simulations for outer PPDs, with ambipolar diffusion and instantaneous cooling, and hence conductive to both instabilities. Given the range of ambipolar Elsässer numbers (Am) explored, it is found that the VSI turbulence dominates over the MRI when ambipolar diffusion is strong (Am = 0.1); the VSI and MRI can co-exist for Am = 1; and the VSI is overwhelmed by the MRI when ambipolar diffusion is weak (Am = 10). Angular momentum transport process is primarily driven by MHD winds, while viscous accretion due to MRI and/or VSI turbulence makes a moderate contribution in most cases. Spontaneous magnetic flux concentration and formation of annular substructures remain robust in strong ambipolar diffusion-dominated discs (Am ≤ 1) with the presence of the VSI. Ambipolar diffusion is the major contributor to the magnetic flux concentration phenomenon rather than advection.

Simulation of vorticity dominated flows using a hybrid approach. II: Numerical method

This thesis presents a numerical study on the interaction between planets and circumstellar disks. We use the hydrodynamics/magnetohydrodynamics code PLUTO(Mignone et al, 2007) to simulate the circustellar accretion disk. A module to include embedded planets was incorporated into the code. We study two critical aspects for planet formation theory: the migration of planets due to gravitational disk torques and the accretion of gas onto planets from the surrounding disk. These two aspects are critical in any planet formation model as they will determine the final mass and the orbital separation. We first investigate aspects for massive planets in the evolutionary phase when a gap has been cleared in the disk. It is found that when a gap has been opened, the migration and gas accretion rate is linearly dependent on the surface density inside the gap. The torques exerted on the planet depend strongly on the material inside the Hill sphere when the local disk mass exceeds the planet mass. The depletion of the Hill sphere due to an accreting planet can increase migration timescales up to an order of magnitude of the linear estimate. Secondly, we investigate migration and gas accretion in turbulent disks, where turbulence is generated by the magnetorotational instability (MRI). In weakly magnetized turbulent disks, low-mass planet migration is dominated by stochastic density perturbations that can be characterized with a given amplitude and correlation time. More massive planets can undergo slower or reversed migration due to unsaturation of the corotation torque by turbulent advection and diffusion of gas into the horseshoe region. Magnetic turbulence is greatly supressed by giant planets that open a gap in the disk. Additionally, Jupiter-mass planets in turbulent disks are found to accrete less than expected from the global-averaged interal stresses in the disk. Our results can be directly implemented in planet population synthesis studies in order to better understand the nature of the observed population of extrasolar planets.

An artificial viscosity, originally designed for Eulerian schemes, is adapted for use in arbitrary Lagrangian–Eulerian simulations. Changes to the Eulerian model (dubbed ‘hyperviscosity’) are discussed, which enable it to work within a Lagrangian framework. New features include a velocity-weighted grid scale and a generalised filtering procedure, applicable to either structured or unstructured grids. The model employs an artificial shear viscosity for treating small-scale vorticity and an artificial bulk viscosity for shock capturing. The model is based on the Navier–Stokes form of the viscous stress tensor, including the diagonal rate-of-expansion tensor. A second-order version of the model is presented, in which Laplacian operators act on the velocity divergence and the grid-weighted strain-rate magnitude to ensure that the velocity field remains smooth at the grid scale. Unlike sound-speed-based artificial viscosities, the hyperviscosity model is compatible with the low Mach number limit. The new model outperforms a commonly used Lagrangian artificial viscosity on a variety of test problems.

view Abstract Citations (25) References (27) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Evolution of Magnetized Protoplanetary Disks Reyes-Ruiz, Mauricio ; Stepinski, Tomasz F. Abstract We investigate the global evolution of a turbulent protoplanetary disk in its viscous stage, incorporating the effects of Maxwell stress due to a large-scale magnetic field permeating disk. We assume that the viscous stress is given by an alpha model. A magnetic field is produced contemporaneously by an alpha omega dynamo mechanism and the resultant Maxwell stress assists the viscous stress in providing the means for disk evolution. The aim of this work is to compare the evolution of magnetized and nonmagnetized disks driven by turbulent viscosity of the same magnitude and thus assess the effects of a self-generated magnetic field on the structure and dynamical evolution of protoplanetary disks. Two illustrative examples corresponding to two different initial conditions are considered: a high-mass case that starts with a disk of 0.245 solar mass and angular momentum of 5.6 x 1052g sq cm/s, and a low-mass that case starts with a disk of 0.11 solar mass and angular momentum of 1.8 x 1052g sq cm/s. For each of these two cases the radial development of a disk is calculated numerically assuming a fiducial value of the dimensionless viscosity parameter alphass = 0.01, as well as alphass = 2 x 10-3. In all cases the central star has a mass equal to 1 solar mass. The most striking feature of magnetized disk evolution is the presence of the surface density bulge located in the region of the disk where the dynamo mechanism cannot support a magnetic field. The bulge persists for a time of the order of 105-106 yr. The presence and persistence of the surface density bulge may have important implications for the process of planet formation and the overall characteristics of resultant planetary systems. Publication: The Astrophysical Journal Pub Date: January 1995 DOI: 10.1086/175120 Bibcode: 1995ApJ...438..750R Keywords: Accretion Disks; Magnetic Fields; Planetary Evolution; Protoplanets; Viscosity; Finite Difference Theory; Ionization; Planetary Systems; Turbulence; Astrophysics; ACCRETION; ACCRETION DISKS; MAGNETOHYDRODYNAMICS: MHD; STARS: PLANETARY SYSTEMS full text sources ADS |

The influence of cross diffusions, Hall and Ion slip of a dissipative magnetized micropolar fluid flow through an infinite concentric rotating vertical cylinders were investigated in addition to the first order chemical reaction. Cylinders are taken into account for Isothermal (constant temperature wall condition) and mixed gradient condition at inner cylinder, while convection cooling and constant wall concentration condition is taken at outer cylinder. The governing equations in cylindrical polar coordinates are coupled ordinary differential equations (ODEs) and are solved numerically with help of Shooting method with fourth order Runge Kutta method. A parametric study illustrating the influence of the emerging parameters on flow, heat and mass transfer components as well as on Skin Friction, Nusselt number and Sherwood number through graphical illustrations.

In this paper, the full nonlinear method based on the three-dimensional potential flow theory and the dynamic analysis method of flexible components are combined to simulate the motion and load response of a FPSO in waves. On the boundary of the hull body, the coupled motions are considered in the impenetrable condition. An improved Eulerian method is adopted to trace strongly nonlinear 3-D free surface deformation. In the far field, artificial damping zone is applied to eliminate reflected wave. Rankine source method is adopted to solve the velocity potential in time domain. Hydrodynamic mesh on hull body is generated by using accumulative chord length cubic parameter spline function. After solving Poisson equation, the initial mesh on free surface becomes orthogonal. In each time step, elastic-mesh-technique (EMT) is used to optimize the mesh on free surface. Several auxiliary functions are introduced to decouple the motions and load, and then the fourth-order Runge-Kutta method is adopted to update the numerical model in time domain. For the forces of the mooring system on the hull at each time step, the dynamic equation of the flexible member is established by the dynamic analysis method based on the elastic slender rod mechanical model, and then the equation is discretized into matrix form by the finite element discretization method and solved. The coupled motion of FPSO hull and mooring system in regular waves of various frequencies in various directions is simulated, and the time domain solutions are obtained. RAO of heave motion in each wave direction is given. The load response at the midship section is analyzed.