ZEUS Code Research Articles

Numerical simulations of wind-driven protoplanetary nebulae. II. signatures of atomic emission

Abstract We follow up on our systematic study of axisymmetric hydrodynamic simulations of protoplanetary nebula. The aim of this work is to generate the atomic analogues of the H2 near-infrared models of Paper I with the ZEUS code modified to include molecular and atomic cooling routines. We investigate stages associated with strong $\mathrm{[Fe\, {\small II}] \, 1.64\, \mathrm{\mu m} }$ and $\mathrm{ [S\, {\small II}] \, 6716}$ Å forbidden lines, the $\mathrm{[O\, {\small I}]\, 6300}$ Å airglow line, and Hα 6563 Å emission. We simulate (80 ∼ 200 km s−1) dense (∼105 cm−3) outflows expanding into a stationary ambient medium. In the case of an atomic wind interacting with an atomic medium, a decelerating advancing turbulent shell thickens with time. This contrasts with all other cases where a shell fragments into a multitude of cometary-shaped protrusions with weak oblique shocks as the main source of gas excitation. We find that the atomic wind-ambient simulation leads to considerably higher excitation, stronger peak and integrated atomic emission as the nebula expands. The weaker emission when one component is molecular is due to the shell fragmentation into fingers so that the shock surface area is increased and oblique shocks are prevalent. Position-velocity diagrams indicate that the atomic-wind model may be most easy to distinguish with more emission at higher radial velocities. With post-AGB winds and shells often highly obscured and the multitude of configurations that are observed, this study suggests and motivates selection criteria for new surveys.

Numerical simulations of wind-driven protoplanetary nebulae – I. near-infrared emission

To understand how the circumstellar environments of post-AGB stars develop into planetary nebulae, we initiate a systematic study of 2D axisymmetric hydrodynamic simulations of protoplanetary nebula (pPN) with a modified ZEUS code. The aim of this first work is to compare the structure of prolate ellipsoidal winds into a stationary ambient medium where both media can be either atomic or molecular. We specifically model the early twin-shock phase which generates a decelerating shell. A thick deformed and turbulent shell grows when an atomic wind expands into an atomic medium. In all other cases, the interaction shell region fragments into radial protrusions due to molecular cooling and chemistry. The resulting fingers eliminate any global slip parallel to the shell surface. This rough surface implies that weak shocks are prominent in the excitation of the gas despite the fast speed of advance. This may explain why low excitation molecular hydrogen is found towards the front of elliptical pPN. We constrain molecular dissociative fractions and timescales of fast $\mathrm H_2$ winds and the pPN lifetime with wind densities $\mathrm{\sim10^{5}cm^{-3}}$ and shock speeds of $\mathrm{80\sim200\,km\,s^{-1}}$. We identify a variety of stages associated with thermal excitation of H$_2$ near-infrared emission. Generated line emission maps and position-velocity diagrams enable a comparison and distinction with post-AGB survey results. The $\mathrm{1\to0 \, S(1)}$ $\&$ $\mathrm{2\to1 \, S(1)}$ lines are lobe-dominated bows rather than bipolar shells.

Shocks in the low angular momentum accretion flow

We address the variability of low luminous galactic nuclei including the Sgr A* or other transient accreting systems, e.g. the black hole X-ray binaries, such as GX 339-4 or IGR J17091. These sources exhibit bright X-ray flares and quasi-periodical oscillations and are theoretically interpreted as the quasi-spherical accretion flows, formed instead of or around Keplerianaccretion disks. In low angular momentum flows the existence of shocks for some range of leading parameters (energy, angular momentum and adiabatic constant of the gas) was studied semi-analytically. The possible hysteresis effect, caused by the fact that the evolution of the flow and the formation of the shock depends on its own history, was discovered. The presence of the shock in the accreted material is important for the observable properties of the out-coming radiation. In the shocked region the gas is dense and hot, thus much more luminous than in the other case. We study the appearance of standing shocks in low angular momentum gas accreting onto a black hole with numerical hydrodynamicalsimulations, using the ZEUS code with Paczynski-Wiitapseudo-Newtonian potential.

ROTATING ACCRETION FLOWS: FROM INFINITY TO THE BLACK HOLE

Accretion onto a supermassive black hole of a rotating inflow is a particularly difficult problem to study because of the wide range of length scales involved. There have been broadly utilized analytic and numerical treatments of the global properties of accretion flows, but detailed numerical simulations are required to address certain critical aspects. We use the ZEUS code to run hydrodynamical simulations of rotating, axisymmetric accretion flows with Bremsstrahlung cooling, considering solutions for which the centrifugal balance radius significantly exceeds the Schwarzschild radius, with and without viscous angular momentum transport. Infalling gas is followed from well beyond the Bondi radius down to the vicinity of the black hole. We produce a continuum of solutions with respect to the single parameter Mdot_Bondi/Mdot_Edd, and there is a sharp transition between two general classes of solutions at an Eddington ratio of Mdot_Bondi/Mdot_Edd ~ few x 10^(-2). Our high inflow solutions are very similar to the standard Shakura & Sunyaev (1973) results. But our low inflow results are to zeroth order the stationary Papaloizou and Pringle (1984) solution, which has no accretion. To next order in the small, assumed viscosity they show circulation, with disk and conical wind outflows almost balancing inflow. These solutions are characterized by hot, vertically extended disks, and net accretion proceeds at an extremely low rate, only of order alpha times the inflow rate. Our simulations have converged with respect to spatial resolution and temporal duration, and they do not depend strongly on our choice of boundary conditions.

RADIAL DEPENDENCE OF PEAK PROTON AND IRON ION FLUXES IN SOLAR ENERGETIC PARTICLE EVENTS: APPLICATION OF THE PATH CODE

The radial dependence of particle peak fluxes in large solar energetic particle (SEP) events is important in determining the potential impact of space weather hazards on space missions. Using the Particle Acceleration and Transport in the Heliosphere code, we model the acceleration and transport of protons and iron ions at evolving coronal mass ejection shocks propagating throughout the inner heliosphere from about 0.1 to 2.5 AU. An example shock with a compression ratio of 3.9 and a speed of 1000 km s{sup -1} close to the Sun is modeled using a two-dimensional MHD ZEUS code. The compression ratio and shock speed weakened to 2.3 and 630 km s{sup -1}, correspondingly, at 2 AU. Shocks with 15 Degree-Sign , 45 Degree-Sign , and 75 Degree-Sign angles between the upstream magnetic field and the shock normal were studied. The shock angle was kept constant throughout the simulation. Both gradual and impulsive events are studied. Diffusive shock acceleration is assumed at the shock and we use a total diffusion coefficient that includes a parallel diffusion coefficient which takes into account the upstream wave amplification, and a perpendicular diffusion coefficient which is based on the NonLinear Guiding Center theory. The transport of particles escapingmore » from the shock is modeled using a Monte Carlo approach. Time-intensity profiles for protons and iron ions are obtained. We analyzed the radial dependence of peak fluxes (J) for both protons and iron ions from 0.5 to 2 AU. We find that the functional dependence is softer than R {sup -3} and is about R {sup -2.9} to R {sup -1.8} in the energy range of 0.3-5 MeV nuc{sup -1}. Quasi-perpendicular shock showed a steeper radial dependence than a quasi-parallel shock. Mixed events show a softer radial dependence at energies above 500 keV nuc{sup -1} for iron ions and above 1 MeV for protons. The values of J(R) depend on seed particle composition, particle energy, shock obliquity, and the interplanetary turbulence level. Consequently, we advocate using SEP event-specific computer modeling rather than empirical formulae for future forecasting of the radiation environment throughout the heliosphere during large SEP events.« less

STAR FORMATION IN A QUASAR DISK

Using a version of the ZEUS code, we carry out two-dimensional simulations of self-gravitating shearing sheets, with application to QSO accretion disks at a few thousand Schwarzschild radii, corresponding to a few hundredths of a parsec for a 10^8 solar-mass black hole. Radiation pressure and optically thick radiative cooling are implemented via vertical averages. We determine dimensionless versions of the maximum surface density, accretion rate, and effective viscosity that can be sustained by density-wave turbulence without fragmentation. Where fragments do form, we study the final masses that result. The maximum Shakura-Sunyaev viscosity parameter is approximately 0.4. Fragmentation occurs when the cooling time is less than about twice the shearing time, as found by Gammie and others, but can also occur at very long cooling times in sheets that are strongly radiation-pressure dominated. For accretion at the Eddington rate onto a 10^8 solar-mass black hole, fragmentation occurs beyond four thousand Schwarzschild radii, r_s. Near this radius, initial fragment masses are several hundred suns, consistent with estimates from linear stability; final masses after merging increase with the size of the sheet, reaching several thousand suns in our largest simulations. With increasing black-hole mass at a fixed Eddington ratio, self-gravity prevails to smaller multiples of r_s, where radiation pressure is more important and the cooling time is longer compared to the dynamical time; nevertheless, fragmentation can occur and produces larger initial fragment masses. We also find energy conservation is likely to be a challenge for all eulerian codes in self-gravitating regimes where radiation pressure dominates.

VISCOUS AND RESISTIVE EFFECTS ON THE MAGNETOROTATIONAL INSTABILITY WITH A NET TOROIDAL FIELD

Resistivity and viscosity have a significant role in establishing the energy levels in turbulence driven by the magnetorotational instability (MRI) in local astrophysical disk models. This study uses the Athena code to characterize the effects of a constant shear viscosity \nu and Ohmic resistivity \eta in unstratified shearing box simulations with a net toroidal magnetic flux. A previous study of shearing boxes with zero net magnetic field performed with the ZEUS code found that turbulence dies out for values of the magnetic Prandtl number, P_m = \nu/\eta, below P_m \sim 1; for P_m \gtrsim 1, time- and volume-averaged stress levels increase with P_m. We repeat these experiments with Athena and obtain consistent results. Next, the influence of viscosity and resistivity on the toroidal field MRI is investigated both for linear growth and for fully-developed turbulence. In the linear regime, a sufficiently large \nu or \eta can prevent MRI growth; P_m itself has little direct influence on growth from linear perturbations. By applying a range of values for \nu and \eta to an initial state consisting of fully developed turbulence in the presence of a background toroidal field, we investigate their effects in the fully nonlinear system. Here, increased viscosity enhances the turbulence, and the turbulence decays only if the resistivity is above a critical value; turbulence can be sustained even when P_m < 1, in contrast to the zero net field model. While we find preliminary evidence that the stress converges to a small range of values when \nu and \eta become small enough, the influence of dissipation terms on MRI-driven turbulence for relatively large \eta and \nu is significant, independent of field geometry.

Vortices in Thin, Compressible, Unmagnetized Disks

We consider the formation and evolution of vortices in a hydrodynamic shearing-sheet model. The evolution is done numerically using a version of the ZEUS code. Consistent with earlier results, an injected vorticity field evolves into a set of long-lived vortices, each of which has a radial extent comparable to the local scale height. But we also find that the resulting velocity field has a positive shear stress, . This effect appears only at high resolution. The transport, which decays with time as t^-1/2, arises primarily because the vortices drive compressive motions. This result suggests a possible mechanism for angular momentum transport in low-ionization disks, with two important caveats: a mechanism must be found to inject vorticity into the disk, and the vortices must not decay rapidly due to three-dimensional instabilities.

Simulating Radiating and Magnetized Flows in Multiple Dimensions with ZEUS‐MP

This paper describes ZEUS-MP, a multi-physics, massively parallel, message-passing implementation of the ZEUS code. ZEUS-MP differs significantly from the thoroughly documented ZEUS-2D code, the completely undocumented (in peer-reviewed literature) ZEUS-3D code, and a marginally documented “version 1” of ZEUS-MP first distributed publicly in 1999. ZEUS-MP offers an MHD algorithm which is better suited for multidimensional flows than the ZEUS-2D module by virtue of modifications to the Method of Characteristics scheme first suggested by Hawley & Stone (1995). This MHD module is shown to compare quite favorably to the TVD scheme described by Ryu et al. (1998). ZEUS-MP is the first publicly-available ZEUS code to allow the advection of multiple chemical (or nuclear) species. Radiation hydrodynamic simulations are enabled via an implicit flux-limited radiation diffusion (FLD) module. The hydrodynamic, MHD, and FLD modules may be used, singly or in concert, in one, two, or three space dimensions. Additionally, so-called “1.5-D” and “2.5-D” grids, in which the “half-D” denotes a symmetry axis along which a constant but non-zero value of velocity or magnetic field is evolved, are supported. Self gravity may be included either through the assumption of a GM/r potential or a solution of Poisson’s equation using one of three linear solver packages (conjugategradient, multigrid, and FFT) provided for that purpose. Point-mass potentials are also supported. Because ZEUS-MP is designed for large simulations on parallel computing platforms, considerable attention is paid to the parallel performance characteristics of each module in the code. Strong-scaling tests involving pure hydrodynamics (with and without self-gravity), MHD, and RHD are performed in which large problems (256 3 zones) are distributed among as many as 1024 processors of an IBM SP3. Parallel efficiency is a strong function of the amount of communication required between processors in a given algorithm, but all modules are shown to scale well on up to 1024 processors for the chosen fixed problem size. Subject headings: hydrodynamics – methods:numerical – methods:parallel – MHD – radiative transfer

Acceleration and transport of heavy ions at coronal mass ejection‐driven shocks

It is now widely accepted that large solar energetic particle (SEP) events are associated with coronal mass ejection (CME)‐driven shocks. Here, particles are often accelerated to approximately MeV energies (and perhaps up to GeV energies) at shock waves driven by CMEs. In this paper, on the basis of our earlier work, we present a detailed model which calculates the acceleration of heavy ions at CME‐driven shocks and their subsequent propagation in the interplanetary medium. The CME‐driven shock is followed numerically using the two‐dimensional ZEUS code. At the shock front and upstream of the shock, the coupled system of streaming protons and stimulated upstream turbulence (assuming the form of Alfvén waves) is considered and the diffusion coefficient of the energetic ions and the stimulated wave intensity is evaluated self‐consistently. The particle diffusion coefficient is then used to determine the maximum momentum of energetic ions. The ion spectra at the shock front are obtained by adopting the steady‐state solution of the transport equation. Across the shock front, the upstream turbulence is amplified and the particle diffusion coefficient is further decreased. Energetic particles in the downstream region convect with the solar wind, cool adiabatically, and diffuse. These effects are followed using a shell model. Particles diffuse ahead of the shock front, can escape and propagate in the interplanetary medium subject to pitch angle scattering. The transport of these particles is followed using a Monte‐Carlo code. Two shocks, one strong and one weak, are considered. Time intensity profiles and particle spectra at various times are obtained for two groups of ions, corresponding to Fe and CNO particles. These results will be helpful on understanding in situ observations at 1 AU made by spacecraft such as ACE and Wind.

On the dynamics of the heliosphere on intermediate and long time-scales

Abstract A time-dependent, three-dimensional model of the dynamics of the heliosphere as a result of solar activity and a time-varying local interstellar medium is presented. The model is based on a recent version of the well known ZEUS code and employs parallel processing. It includes the solar and interstellar plasma components as well as neutral atoms, and contains the heliospheric magnetic field in a kinematic fashion. We study the dynamics of the heliosphere due to solar activity on periods of months to years up to the so-called Schwabe (11-year) cycle as well as due to time variations of the local interstellar medium, all of which have drawn increasing attention during recent years, as the significance of their direct or indirect effect on the Earth and its environment is under lively debate.

Numeric simulations of in situ observations of interacting ejecta near 1 AU

Abstract We use a simple numerical model (Gonzalez-Esparza, J.A., Santillan, A., Ferrer, J. A numerical study of the interaction between two ejecta in the interplanetary medium: one and two dimensional hydrodynamic simulations, Ann. Geophys. 22, 3741–3749, 2004) to study the evolution of three events in the solar wind reported by Wang et al. (Wang, Y.M., Ye, P.Z., Wang, S. Multiple magnetic clouds: several examples during March–April 2001. J. Geophys. Res. 108, 1370, 2003, doi:10.1029/2003JA009850 ), where two interacting ejecta detected in situ by ACE near 1 AU were related to CMEs observed previously by SOHO-LASCO. The study is based on a 1-D hydrodynamic model using the ZEUS code (Stone, J.M., Norman, M. ZEUS 2-D: A radiation magnetohydrodynamics code for astrophysical flows in two dimensions, I, the hydrodynamics algorithms and tests, Astrophys. J. 80, 753, 1992). Although this model cannot address either the magnetic field dynamics or the complex geometrical effects intrinsic in the three-dimensional nature of the phenomena, it illuminates the transferring of momentum and evolution of interacting large-scale solar wind disturbances in those cases where there is no merging (magnetic reconnection) between the two ejecta. This model can reproduce, in some cases, characteristics of the events such as transit times and flow signatures as inferred from the two-point measurements from spacecraft.

ON THE GALACTIC SPIRAL PATTERNS: STELLAR AND GASEOUS

The gas response to a proposed spiral stellar pattern for our Galaxy is presented here as calculated via 2D hydrodynamic calculations utilizing the ZEUS code in the disk plane. The locus is that found by Drimmel (2000) from emission profiles in the K band and at 240 <TEX>${\mu}m$</TEX>. The self-consistency of the stellar spiral pattern was studied in previous work (see Martos et al. 2004). It is a sensitive function of the pattern rotation speed, <TEX>$\Omega$</TEX>p, among other parameters which include the mass in the spiral and its pitch angle. Here we further discuss the complex gaseous response found there for plausible values of <TEX>$\Omega$</TEX>p in our Galaxy, and argue that its value must be close to <TEX>$20 km s^{-l}\;kpc^{-1}$</TEX> from the strong self-consistency criterion and other recent, independent studies which depend on such parameter. However, other values of <TEX>$\Omega$</TEX>p that have been used in the literature are explored to study the gas response to the stellar (K band) 2-armed pattern. For our best fit values, the gaseous response to the 2-armed pattern displayed in the K band is a four-armed pattern with complex features in the interarm regions. This response resembles the optical arms observed in the Milky Way and other galaxies with the smooth underlying two-armed pattern of the old stellar disk populations in our interpretation. The complex gaseous response appears to be related to resonances in stellar orbits. Among them, the 4:1 resonance is paramount for the axisymmetric Galactic model employed, and the set of parameters explored. In the regime seemingly proper to our Galaxy, the spiral forcing appears to be marginally strong in the sense that the 4:1 resonance terminates the stellar pattern, despite its relatively low amplitude. In current work underway, the response for low values of <TEX>$\Omega$</TEX>p tends to remove most of the rich structure found for the optimal self-consistent model and the gaseous pattern is ring-like. For higher values than the optimal, more features and a multi-arm structure appears.

A plausible Galactic spiral pattern and its rotation speed

We report calculations of the stellar and gaseous response to a Milky Way mass distribution model including a spiral pattern with a locus as traced by K-band observations, over imposed on the axisymmetric components in the plane of the disk. The stellar study extends calculations from previous work concerning the self-consistency of the pattern. The stellar response to the imposed spiral mass is studied via computations of the central family of periodic and nearby orbits as a function of the pattern rotation speed, $\Omega_p$, among other parameters. A fine grid of values of $\Omega_p$ was explored ranging from 12 to 25 $km s^{-1} kpc^{-1}$. Dynamical self-consistency is highly sensitive to $\Omega_p$, with the best fit appearing at 20 $km s^{-1} kpc^{-1}$. We give an account of recent independent pieces of theoretical and observational work that are dependent on the value of $\Omega_p$, all of which are consistent with the value found here; the recent star formation history of the Milky Way, local inferences of cosmic ray flux variations and Galactic abundance patterns. The gaseous response, which is also a function of $\Omega_p$, was calculated via 2D hydrodynamic simulations with the ZEUS code. For $\Omega_p = 20 km s^{-1} kpc^{-1}$, the response to a two-armed pattern is a structured pattern of 4 arms, with bifurcations along the arms and interarm features. The pattern resembles qualitatively the optical arms observed in our Galaxy and other galaxies. The complex gaseous pattern appears to be linked to resonances in stellar orbits. Among these, the 4:1 resonance plays an important role, as it also determines the extent over which the imposed spiral pattern is dynamically self-consistent.

Numerical Models of Viscous Accretion Flows near Black Holes

We report on a numerical study of viscous fluid accretion onto a black hole. The flow is axisymmetric and uses a pseudo-Newtonian potential to model relativistic effects near the event horizon. The numerical method is a variant of the ZEUS code. As a test of our numerical scheme, we are able to reproduce results from earlier, similar work by Igumenshchev and Abramowicz and Stone et al. We consider models in which mass is injected onto the grid as well as models in which an initial equilibrium torus is accreted. In each model we measure three ``eigenvalues'' of the flow: the accretion rate of mass, angular momentum, and energy. We find that the eigenvalues are sensitive to r_{in}, the location of the inner radial boundary. Only when the flow is always supersonic on the inner boundary are the eigenvalues insensitive to small changes in r_{in}. We also report on the sensitivity of the results to other numerical parameters.

The ZEUS code for astrophysical magnetohydrodynamics: new extensions and applications

A brief review of the numerical methods implemented in the widely used ZEUS code for astrophysical magnetohydrodynamics (MHD) is given. Extensions of the numerical methods to treat problems in nonideal MHD in a variety of regimes are discussed. In particular, methods for treating Ohmic dissipation, and methods for studying partially ionized plasmas in which the ion and neutral components are weakly coupled through a collisional drag term are considered. Recent application of the methods to the study of the dynamics of accretion disks is described.

Modelling of magnetic interactions in partially-ionized gas: application to the FIP effect

We have adapted the ZEUS code to model magnetic interactions in partially ionized gas. When two regions of opposite polarity come into contact with each other, ions drifting in response to the Lorentz force fall into the minimum of the magnetic field, and then the drifting ions force the neutrals to take part in the flow. Because of the finite time required for ion-atom collisions to occur, the gas which emerges from the interaction site has an ion/atom ratio which may be altered relative to that in the ambient medium. In order to model this effect, we adapt the Zeus code to a two-step iterative process involving a cycle between the hydrodynamic (HD) and the magnetohydrodynamic (MHD) versions of the code. The ion and atom fluids are coupled by collisions. Our simulations show that in chromospheric conditions, outflowing gas exhibits enhancements in ion/atom ratios which may be as large as a factor of 10 or more. The magnitude of the enhancements is determined by two key ratios which enter into the problem: the degree of ionization (ni/na), and the plasma β parameter. We show that, in the context of the mechanism we propose here, the amplitude of the ion/atom enhancements in the solar chromosphere is subject to a remarkable self-regulation because the ion density ni is almost invariant over the height range of interest to us. Our results are relevant in the context of the Sun, where the coronal abundances of elements with low first ionization potential (FIP) are systematically enhanced in certain magnetic structures. Although data for stars other than the Sun are sparse, we point out that our results are also useful for interpreting the available stellar data.

Nonlinear Development and Observational Consequences of Wardle C‐Shock Instabilities

We compute the nonlinear development of the instabilities in C-shocks first described by Wardle, using a version of the ZEUS code modified to include a semi-implicit treatment of ambipolar diffusion. We find that, in three dimensions, thin sheets form parallel to the shock velocity and perpendicular to the magnetic field lines. High-resolution, two-dimensional models show that the sheets are confined by the Brandenburg & Zweibel ambipolar diffusion singularity, forcing them to numerically unresolvable thinness. Hot and cold regions form around these filaments, disrupting the uniform temperature structure characteristic of a steady-state C-shock. This filamentary region steadily grows as the shock progresses. We compare steady-state to unstable C-shocks, showing excitation diagrams, images, line ratios, and line profiles for molecular hydrogen lines visible in the K band with the Infrared Space Observatory and with NICMOS on the Hubble Space Telescope.

Numerical modeling of hypervelocity impact phenomena with desktop computers

This paper discusses recent developments in personal computer based software for numerical simulation of impact phenomena, with emphasis on hypervelocity impact. The essential features of the ZeuS code are discussed, as a representative example of computer codes implemented on microcomputers. Simulations of hypervelocity impact phenomena involving debris bubble formation in thin plate impacts and crater growth in thick plate impacts are presented. Agreement between calculations and experimental results is excellent for situations where data is available.

Induced roll computations for conventional missiles

The ZEUS program, which solves Euler's equations by spatial marching, is applied to conventional missiles characterized by circular cross-sectional bodies, canard control surfaces, and a cruciform tail. ZEUS incorporates a multiple-zone gridding technique and a second-order extension of Godunov's method. The Godunov method is an upwind scheme based on the Riemann problem for steady supersonic flow and cast in a control volume form. Second-order accuracy is obtained by computing local slopes and adding a predictor step. The scheme fits the bow shock and captures imbedded shocks. The ZEUS code is applied to two canard-controlled missiles with cruciform tails to predict longitudinal and lateral-directional (e.g., induced rolling moment) aerodynamic characteristics.