Axisymmetric Disk Research Articles

On the Dynamics of the Inclination Instability.

Axisymmetric disks of eccentric Kepler orbits are vulnerable to an instability that causes orbits to exponentially grow in inclination, decrease in eccentricity, and cluster in their angle of pericenter. Geometrically, the disk expands to a cone shape that is asymmetric about the mid-plane. In this paper, we describe how secular gravitational torques between individual orbits drive this "inclination instability". We derive growth timescales for a simple two-orbit model using a Gauss N-ring code, and generalize our result to larger N systems with N-body simulations. We find that two-body relaxation slows the growth of the instability at low N and that angular phase coverage of orbits in the disk is important at higher N. As N → ∞, the e-folding timescale converges to that expected from secular theory.

Surface waves in protoplanetary disks induced by outbursts: Concentric rings in scattered light

Context. Vertically hydrostatic protoplanetary disk models are based on the assumption that the main heating source, stellar irradiation, does not vary much with time. However, it is known that accreting young stars are variable sources of radiation. This is particularly evident for outbursting sources such as EX Lupi and FU Orionis stars. Aims. We investigate how such outbursts affect the vertical structure of the outer regions of the protoplanetary disk, in particular their appearance in scattered light at optical and near-infrared wavelengths. Methods. We employ the 3D FARGOCA radiation-hydrodynamics code, in polar coordinates, to compute the time-dependent behavior of the axisymmetric disk structure. The temperature is computed self-consistently and time-dependently from the irradiation flux using a two-stage radiative transfer method: first the direct illumination is computed; then the diffuse radiation is treated with the flux-limited diffusion method. The outbursting inner disk region is not included explicitly. Instead, its luminosity is added to the stellar luminosity and is thus included in the irradiation of the outer disk regions. For time snapshots of interest we insert the density structure into the RADMC-3D radiative transfer code and compute the appearance of the disk at optical/near-infrared wavelengths, where we observe stellar light that is scattered off the surface of the disk. Results. We find that, depending on the amplitude of the outbursts, the vertical structure of the disk can become highly dynamic, featuring circular surface waves of considerable amplitude. These “hills” and “valleys” on the disk’s surface show up in the scattered light images as bright and dark concentric rings. Initially these rings are small and act as standing waves, but they subsequently lead to outward propagating waves, like the waves produced by a stone thrown into a pond. These waves continue long after the actual outburst has died out. Conclusions. Single, periodic, or quasiperiodic outbursts of the innermost regions of protoplanetary disks will necessarily lead to wavy structures on the surface of these disks at all radii. We propose that some of the multi-ringed structures seen in optical/infrared images of several protoplanetary disks may have their origin in outbursts that occurred decades or centuries ago. However, the multiple rings seen at (sub-)millimeter wavelengths in HL Tau and several other disks are not expected to be related to such waves.

Parametric Study of the Rossby Wave Instability in a Two-dimensional Barotropic Disk. II. Nonlinear Calculations

Abstract Vortices in protoplanetary disks have attracted attention since the discovery of lopsided structures. One of the possible mechanisms for producing vortices is the Rossby wave instability (RWI). In our previous work, we have performed detailed linear stability analyses of the RWI with various initial conditions. In this paper, we perform numerical simulations of the vortex formation by the RWI in two-dimensional barotropic disks using the Athena++ code. As initial conditions, we consider axisymmetric disks with a Gaussian surface density bump of various contrasts and half-widths. Perturbations grow as expected from the linear stability analyses in the linear and weakly nonlinear regimes. After the saturation, multiple vortices are formed in accordance with the most unstable azimuthal mode and coalesce one after another. In the end, only one quasi-stationary vortex (the RWI vortex) remains, which migrates inward. During the RWI evolution, the axisymmetric component approaches the stable configuration. We find that the axisymmetric component reaches the marginally stable state for the most unstable azimuthal mode at the saturation and for the m = 1 mode at the final vortex merger. We investigate the structure and evolution of the RWI vortices. We obtain some empirical relations between the properties of the RWI vortices and the initial conditions. Using tracer particle analyses, we find that the RWI vortex can be considered as a physical entity, like a large fluid particle. Our results provide solid theoretical ground for quantitative interpretation of the observed lopsided structures in protoplanetary disks.

On the non-reacting flow and mixing fields of an axisymmetric disk stabilizer, under inlet mixture stratification and preheat

The non-reacting, time mean and fluctuating flow field and the mixing characteristics of a bluff-body stratified flame stabilizer have been investigated for a range of inlet mixture compositions and preheat conditions. The burner consists of three concentric disks that form two premixing cavities. Tracer fuel is injected in the first cavity and is partially premixed with air flowing through the cavity system, resulting in a stratified equivalence ratio profile at the inlet of the stabilization region. The burner is capable of anchoring flames at very lean mixtures, down to Φ = 0.13 at 743 K. Particle Image Velocimetry (PIV) and Fourier Transform Infrared Spectroscopy (FTIR) analysis have been performed to evaluate the flow and mixing fields developing in the downstream near wake. Four levels of preheat of the incoming reactants, ranging from 300 to 743 K, for lean and ultra-lean, limiting, mixture levels have been selected. The impact of preheat on the developing disk wake topology and mixing performance has been examined and its operational parameters have been evaluated. Further, the study has helped to elucidate the effects of inlet mixture stratification, alone or in combination with preheat, on the performance characteristics of the disk stabilizer and to identify parameters that control mixture placement in the recirculation zone.

Numerical computation of gravitational field of infinitely-thin axisymmetric disc with arbitrary surface mass density profile and its application to preliminary study of rotation curve of M33

AbstractWe developed a numerical method[-70pt] to compute the gravitational field of an infinitely-thin axisymmetric disc with an arbitrary surface mass density profile. We evaluate the gravitational potential by a split quadrature using the double exponential rule and obtain the acceleration vector by numerically differentiating the potential by Ridders’ algorithm. By using the new method, we show the rotation curves of some non-trivial discs: (i) truncated power-law discs, (ii) discs with a non-negligible center hole, (iii) truncated Mestel discs with edge-softening, (iv) double power-law discs, (v) exponentially-damped power-law discs, and (vi) an exponential disc with a sinusoidal modulation of the density profile. Also, we present a couple of model fittings to the observed rotation curve of M33: (i) the standard deconvolution by assuming a spherical distributin of the dark matter and (ii) a direct fit of infinitely-thin disc mass with a double power-law distribution of the surface mass density.

A Magnetic Field Connecting the Galactic Center Circumnuclear Disk with Streamers and Mini-spiral: Implications from 850 μm Polarization Data

Abstract Utilizing James Clark Maxwell Telescope 850 μm SCUPOL dust polarization data, we investigate the configuration of the magnetic (B) field in the circumnuclear disk (CND) of the Galactic center. The SCUPOL data show a highly improved polarization coverage and resolution compared with earlier 100 μm observations. The 850 μm data have a resolution and coverage similar to previous 350 μm polarimetry data. However, with a proper sampling on a 10″ grid, we find that the 850 μm data trace the morphological structures of the CND substantially better. Furthermore, because the 850 μm trace the field deeper into the material near SgrA*, they represent the highest-resolution submillimeter probe to date of the CND magnetic field. The observed B-field morphology is well described by a self-similar axisymmetric disk model where the radial infall velocity is one-quarter of the rotational velocity. A detailed comparison with higher-resolution interferometric maps from the Submillimeter Array further reveals that the B-field aligns with the neutral gas streamers connecting to the CND. Moreover, the innermost observed B-field structure also appears to trace and align with the mini-spiral located inside the CND. This suggests that there is one underlying B-field structure that is connecting the CND with its streamers and the inner mini-spiral. An estimate of β Plasma ≲ 1—based on the global B-field morphology that constrains the azimuthal-to-vertical field strength ratio of around 40 combined with a measurement of the azimuthal velocity indicates that the B-field appears dynamically significant toward the CND and also onwards to the inner mini-spiral.

Acoustic radiation torque of an acoustic-vortex spanner exerted on axisymmetric objects

Based on the analysis of the wave vector of an acoustic-vortex (AV) spanner, the radiation torque of object rotation is investigated. It is demonstrated that the rotation of an axisymmetric disk centered on the AV spanner is mainly driven by the acoustic radiation force. The radiation torque exerted on a small-radius object is inversely associated with the topological charge in the center AV, and it is enhanced significantly for a larger AV with a higher topological charge. With the sixteen-source experimental setup, radius dependencies of radiation torque for AV spanners with different topological charges are verified by quantitative laser-displacement measurements using disks with different radii. The favorable results demonstrate that the radiation torque is more applicable than the orbital angular momentum in describing the driving capability of an AV spanner and can be used as an effective tool in clinical applications to manipulate objects with a feature size at the wavelength-scale inside body.

Lift forces on axial symmetry particles rotating in a linear shear flow of a rarefied gas

We theoretically investigate the lift forces (including shear lift and rotational lift) experienced by nonspherical convex particles rotating in a linear shear flow of a rarefied gas, based on the gas kinetic theory. In our model, the interactions between the gas molecules and the particle surface are assumed to be the Maxwell combined specular-diffuse scattering. By integrating the general expressions for several simple axisymmetric bodies, including spheres, cylinders, disks, and spheroids, the specific expressions for the lift forces acting on these bodies are obtained. The pitching effect of the nonspherical particles results in a transverse force for the shear lift, but there is no transverse force for the rotational lift. The expressions for the orientation-averaged shear lift forces are obtained under a uniform distribution of the particle orientation.

Combined dynamical effects of the bar and spiral arms in a Galaxy model. Application to the solar neighbourhood

Context. Observational data indicate that the Milky Way is a barred spiral galaxy. Computation facilities and availability of data from Galactic surveys stimulate the appearance of models of the Galactic structure, however further efforts are needed to build dynamical models containing both spiral arms and the central bar/bulge. Aims. We expand the study of the stellar dynamics in the Galaxy by adding the bar/bulge component to a model with spiral arms introduced in one of our previous publications. The model is tested by applying it to the solar neighbourhood, where observational data are more precise. Methods. We model analytically the potential of the Galaxy to derive the force field in its equatorial plane. The model comprises an axisymmetric disc derived from the observed rotation curve, four spiral arms with Gaussian-shaped groove profiles, and a classical elongated/oblate ellipsoidal bar/bulge structure. The parameters describing the bar/bulge are constrained by observations and the stellar dynamics, and their possible limits are determined. Results. A basic model results in a bar of 2.9 kpc in length, with a mass of the order of a few 109M⊙ (which does not include the axisymmetric part of the bulge, which has a mass of about 1010M⊙). The size and orientation of the bar are also restricted by the position of masers with Very Long Baseline Interferometry (VLBI). The bar’s rotation speed is constrained to Ωbar < 50 km s−1 kpc−1 taking into account the allowed mass range. Conclusions. We conclude that our basic model is compatible with observations and with the dynamical constraints. The model explains simultaneously the bulk of the main moving groups, associated here with the spiral corotation resonance, and the Hercules stream, associated with several inner high-order spiral resonances; in particular, with the 8/1 resonance. From the dynamical constraints on the bar’s angular speed, it is unlikely that the bar’s outer Lindblad resonance (OLR) lies near the solar circle; moreover, its proximity would compromise the stability of the local arm structure.

3D dynamic coupled thermoelastic solution for constant thickness disks using refined 1D finite element models

This paper deals with the generalized coupled thermoelastic solution for disks with constant thickness. It is a sequel to the authors’s previous work in which refined 1D Galerkin finite element models with 3D-like accuracies are developed for theories of coupled thermoelasticity. Use of the reduced models with low computational costs may be of interest in a laborious time history analysis of the dynamic problems. In this paper, the developed models are applied and evaluated for a 3D solution of the dynamic generalized coupled thermoelasticity problem in the disk subjected to thermal shock loads. Comparison of the obtained result with the results available in the literature verified the proposed finite element models are quite efficient with very high rate of convergence and able to provide results with analytical accuracy. In addition, propagation of the thermoelastic waves, the wave reflection from the boundaries and the Poisson effect in an axisymmetric and asymmetric disk problem are represented as contour plots to demonstrate 3D capabilities of the models.

Moving mode shape function approach for spinning disk and asymmetric disc brake squeal

The solution approach of an asymmetric spinning disk under stationary friction loads requires the mode shape function fixed in the disk in the assumed mode method when the equations of motion is described in the space-fixed frame. This model description will be termed the 'moving mode shape function approach' and it allows us to formulate the stationary contact load problem in both the axisymmetric and asymmetric disk cases.Numerical results show that the eigenvalues of the time-periodic axisymmetric disk system are time-invariant. When the axisymmetry of the disk is broken, the positive real parts of the eigenvalues highly vary with the rotation of the disk in the slow speeds in such application as disc brake squeal. By using the Floquet stability analysis, it is also shown that breaking the axisymmetry of the disc alters the stability boundaries of the system.

The secular evolution of discrete quasi-Keplerian systems

A discrete self-gravitating quasi-Keplerian razor-thin axisymmetric stellar disc orbiting a massive black hole sees its orbital structure diffuse on secular timescales as a result of a self-induced resonant relaxation. In the absence of collective effects, such a process is described by the recently derived inhomogeneous multi-mass degenerate Landau equation. Relying on Gauss’ method, we computed the associated drift and diffusion coefficients to characterise the properties of the resonant relaxation of razor-thin discs. For a disc-like configuration in our Galactic centre, we showed how this secular diffusion induces an adiabatic distortion of orbits and estimate the typical timescale of resonant relaxation. When considering a disc composed of multiple masses similarly distributed, we have illustrated how the population of lighter stars will gain eccentricity, driving it closer to the central black hole, provided the distribution function increases with angular momentum. The kinetic equation recovers as well the quenching of the resonant diffusion of a test star in the vicinity of the black hole (the “Schwarzschild barrier”) as a result of the divergence of the relativistic precessions. The dual stochastic Langevin formulation yields consistent results and offers a versatile framework in which to incorporate other stochastic processes.

Viscous Overstability in Saturn’s Rings: Influence of Collective Self-gravity

Abstract We investigate the influence of collective self-gravity forces on the nonlinear, large-scale evolution of the viscous overstability in Saturn’s rings. We numerically solve the axisymmetric nonlinear hydrodynamic equations in the isothermal and non-isothermal approximation, including radial self-gravity and employing transport coefficients derived by Salo et al. We assume optical depths to model Saturn’s dense rings. Furthermore, local N-body simulations, incorporating vertical and radial collective self-gravity, are performed. Vertical self-gravity is mimicked through an increased frequency of vertical oscillations, while radial self-gravity is approximated by solving the Poisson equation for an axisymmetric thin disk with a Fourier method. Direct particle–particle forces are omitted, which prevents small-scale gravitational instabilities (self-gravity wakes) from forming, an approximation that allows us to study long radial scales and to compare directly the hydrodynamic model and the N-body simulations. Our isothermal and non-isothermal hydrodynamic model results with vanishing self-gravity compare very well with results of Latter & Ogilvie and Rein & Latter, respectively. In contrast, for rings with radial self-gravity we find that the wavelengths of saturated overstable waves settle close to the frequency minimum of the nonlinear dispersion relation, i.e., close to a state of vanishing group velocities of the waves. Good agreement is found between non-isothermal hydrodynamics and N-body simulations for moderate and strong radial self-gravity, while the largest deviations occur for weak self-gravity. The resulting saturation wavelengths of viscous overstability for moderate and strong self-gravity ( ) agree reasonably well with the length scales of axisymmetric periodic microstructure in Saturn’s inner A ring and the B ring, as found by Cassini.

Spatially varying heat flux driven close-contact melting – A Lagrangian approach

Close-contact melting refers to the process of a heat source melting its way into a phase change material. Of special interest is the close-contact melting velocity, or more specifically the relative velocity between the heat source and the phase change material. In this work, we present a novel numerical approach to simulate quasi-steady, heat flux driven close-contact melting. It extends existing approaches found in the literature, and, for the first time, allows to study the impact of a spatially varying heat flux distribution. We will start by deriving the governing equations in a Lagrangian reference frame fixed to the heat source. Exploiting the narrowness of the melt film enables us to reduce the momentum balance to the Reynolds equation, which is coupled to the energy balance via the velocity field. We particularize our derivation for two simple, yet technically relevant geometries, namely an axi-symmetric circular disc and a 2d planar heat source. An iterative solution procedure for the coupled system is described in detail and discussed on the basis of a convergence study. Furthermore, we present an extension to allow for rotational melting modes. Various test cases demonstrate the proficiency of our method. In particular, we will utilize the method to assess the efficiency of the close-contact melting process and to quantify the model error introduced if convective losses are neglected. Finally, we will draw conclusions and present an outlook to future work.

Signatures of hypermassive neutron star lifetimes on r-process nucleosynthesis in the disc ejecta from neutron star mergers

We investigate the nucleosynthesis of heavy elements in the winds ejected by accretion disks formed in neutron star mergers. We compute the element formation in disk outflows from hypermassive neutron star (HMNS) remnants of variable lifetime, including the effect of angular momentum transport in the disk evolution. We employ long-term axisymmetric hydrodynamic disk simulations to model the ejecta, and compute r-process nucleosynthesis with tracer particles using a nuclear reaction network containing $\sim 8000$ species. We find that the previously known strong correlation between HMNS lifetime, ejected mass, and average electron fraction in the outflow is directly related to the amount of neutrino irradiation on the disk, which dominates mass ejection at early times in the form of a neutrino-driven wind. Production of lanthanides and actinides saturates at short HMNS lifetimes ($\lesssim 10$ ms), with additional ejecta contributing to a blue optical kilonova component for longer-lived HMNSs. We find good agreement between the abundances from the disk outflow alone and the solar r-process distribution only for short HMNS lifetimes ($\lesssim 10$ ms). For longer lifetimes, the rare-earth and third r-process peaks are significantly under-produced compared to the solar pattern, requiring additional contributions from the dynamical ejecta. The nucleosynthesis signature from a spinning black hole (BH) can only overlap with that from a HMNS of moderate lifetime ($\lesssim 60$ ms). Finally, we show that angular momentum transport not only contributes with a late-time outflow component, but that it also enhances the neutrino-driven component by moving material to shallower regions of the gravitational potential, in addition to providing additional heating.

The self-electric field effect on the MRI instability of magnetized rotational flows: Cylindrical model

The self-electric field effect on the magnetorotational instability (MRI) of non-neutral rotating electron-ion plasmas, related to axisymmetric disks, is investigated by using a cylindrical model. Describing the equilibrium state of the rotational system in the presence of a self-electric field, the fluid-Maxwell equations as well as the linear perturbation theory are employed to derive the dispersion relation (DR) of the excited modes in the system. The obtained DR analysis leads to analytical stability conditions, the growth rate of the instability, and the saturation amount of the pure instability excited in the system, depending on the equilibrium velocity difference between the involved components. Moreover, the numerical analysis of the obtained DR shows that the self-electric field has an important role for the MRI in the region where the angular frequency due to drift, , dominates over all the frequencies, in the outer part of non-neutral disks. In this region, the growth rate of the MRI increases dramatically with the ratio of the ion density to electron density until it saturates. On the other hand, in the inner region of the disk in which the gravitational frequency overcomes the others, the growth rate of the instability is found to be much smaller than that of the outer region. These results can be applicable to understand the MRI growth rate of the differentially rotating non-neutral plasma systems existing in the astrophysical environments, particularly in the magnetosphere of a neutron star.

Rapid distortion of turbulence into an open turbine rotor

Rapid distortion of turbulence (RDT) theory is applied to homogeneous, isotropic turbulence incident on a horizontal axis turbine rotor such as a wind turbine or tidal-stream turbine. The mean flow field of the rotor which distorts the turbulence is represented by the commonly used axisymmetric actuator disc model due to Betz and Joukowski. The fluctuating streamwise component of the turbulence distorted by this field is calculated at the actuator disc plane. Turbulence velocity intensities and spectra are evaluated for general ratios of turbulence integral length scale to the rotor diameter, including the small-scale limit for which the original homogeneous strain analysis of Batchelor and Proudman may be applied. The distortion of the mean velocity profile of an incident rotor wake which may be considered a zero frequency disturbance relevant to wind and tidal turbine operation in large arrays is also analysed by the same method, treating it as a deterministic disturbance in the incident flow.

Finite Volume, Computational Fluid Dynamics-Based Investigation of Supercavity Pulsations

Computations of pulsating supercavity flows behind axisymmetric disk cavitators are presented. The method of computation is a finite volume discretization of the equations of mixture fluid motion. The gas phase is treated as compressible, the liquid phase as incompressible, and the interface accuracy enhanced using a volume of fluid (VOF) approach. The re-entrant, pulsating, and twin vortex modes of cavity closure are delineated and computationally resolved, including the expected hysteresis. A phase diagram of cavitation number versus ventilation rate at three Froude conditions is computationally constructed. Sample re-entrant, pulsation, and twin vortex snapshots are presented. Pulsation results are compared with stability criterion from the literature as well as examined for their expected character. Computations appear to capture the complete spectrum of cavity closure conditions. A detailed comparison of computational simulation and physical experiment at similar conditions is also included as a means to validate the computational results.

Radial Surface Density Profiles of Gas and Dust in the Debris Disk around 49 Ceti

Abstract We present ∼0.″4 resolution images of CO(3–2) and associated continuum emission from the gas-bearing debris disk around the nearby A star 49 Ceti, observed with the Atacama Large Millimeter/Submillimeter Array (ALMA). We analyze the ALMA visibilities in tandem with the broadband spectral energy distribution to measure the radial surface density profiles of dust and gas emission from the system. The dust surface density decreases with radius between ∼100 and 310 au, with a marginally significant enhancement of surface density at a radius of ∼110 au. The SED requires an inner disk of small grains in addition to the outer disk of larger grains resolved by ALMA. The gas disk exhibits a surface density profile that increases with radius, contrary to most previous spatially resolved observations of circumstellar gas disks. While ∼80% of the CO flux is well described by an axisymmetric power-law disk in Keplerian rotation about the central star, residuals at ∼20% of the peak flux exhibit a departure from axisymmetry suggestive of spiral arms or a warp in the gas disk. The radial extent of the gas disk (∼220 au) is smaller than that of the dust disk (∼300 au), consistent with recent observations of other gas-bearing debris disks. While there are so far only three broad debris disks with well characterized radial dust profiles at millimeter wavelengths, 49 Ceti’s disk shows a markedly different structure from two radially resolved gas-poor debris disks, implying that the physical processes generating and sculpting the gas and dust are fundamentally different.

Amplified effect of mild plastic anisotropy on residual stress and strain anisotropy

Axisymmetric indentation of a geometrically axisymmetric disk produced residual stresses by non-uniform plastic deformation. The 2024 aluminum plate used to make the disk exhibited mild plastic anisotropy with about 10% lower strength in the transverse direction compared to the rolling and through-thickness directions. Residual stresses and strains in the disk were measured with neutron diffraction, slitting, the contour method, x-ray diffraction and hole drilling. Surprisingly, the residual-stress anisotropy measured in the disk was about 40%, the residual-strain anisotropy was an impressive 100%, and the residual stresses were higher in the weaker direction. The high residual stress anisotropy relative to the mild plastic anisotropy and the direction of the highest stress are explained by considering the mechanics of indentation: constraint on deformation provided by the material surrounding the indentation and preferential deformation in the most compliant direction for incremental deformation. By contrast, the much larger anisotropy in residual strain compared to that in residual stress is independent of the fabrication process and is instead explained by considering Hookean elasticity. For Poisson's ratio of 1/3, the relationship simplifies to the residual strain anisotropy equaling the square of the residual stress anisotropy, which matches the observed results (2 ≈ 1.4^2). A lesson from this study is that to accurately predict residual stresses and strains, one must be wary of seemingly reasonable simplifying assumptions such as neglecting mild plastic anisotropy.