Numerics

Truelove Klein Mc Kee et al. (‘98) - AMR

​Self-Gravitational hydrodynamics with 3d AMR: methodolgy and applications to molecular cloud collapse and fragmentation.

Summary

Develops methods for AMR, dynamic grids that allot finer resolution over many length scales that is imperative for studying problems of gravitational collapse. The criteria for refinement is crucial, and it is the jeans condition. This sets the resolution smaller than the local Jean’s condition, allowing new benchmarks in the probing of the dynamics. They find uniformly rotating spherical clouds to collapse along the equatorial plane. When perturbed, these form ‘filamentary singularties’ that don’t fragment when isothermal.

Methodology

This paper goes through in extensive detail on the 3 components of their code methodology: the hyperbolic solvers that employ the Gudonov method for solution of the hydro equations, elliptic solvers that utilize AMR multigrid method to solve Poisson’s equation, and finally these two methods operation within an AMR framework. The use of stencils as different layers of cell-centered quantities used for averaging the node centered quantities (for self - gravity) is detailed. Paper refers to Almgren for discussion on AMR multigrid cycle procedures.

AMR and refinement criteria

Paper refers to Pember et al. (‘98) for a summary of the procedure for advancing grids. Their methods are based on Berger and Oliger (‘84), Berger and Collela (‘89), and Bell et al. (‘94).

To trigger refinement from level 0 to level 1, the code employs a density criterion to cells that contain gas of the original cloud. Such cells are identified if they contain a density greater than or equal to ½ the original cloud density at the outer edge. This results in the level 0 cells effectively removing the boundary of the computational volume from the surface of the cloud. Jean's condition is than used for refinement to 2nd and higher levels (discussed below).

The Truelove criteria

Jeans' analysis of the linearized 1D gravitohydrodynamic (GHD) equations of a medium of infinite extent lead to the expression for the jeans length:

\lambda_j = (\frac{\pi c_s^2}{G \rho})^{1/2}$

which showed that perturbations that are larger than this are gravitationally unstable and collapse. Truelove ('97) showed that these instabilities can be triggered from numerical errors of the GHD solvers. Errors introduced on the cell scale at coarser grids can be transmitted to finer levels and can lead to 'artificial fragmentation'. A way to avoid this is to maintain resolution of the local jeans length. By defining:

J = \frac{\triangle x}{\lambda_j} $

Truelove ('97) showed that keeping

J \leqq 0.25$

prevented artificial fragmentation of an isothermal cloud spanning 7 decades of density. It is expected to be necessary albeit not necessarily sufficient for isothermal collapse. By not having the resolution of gradients can trigger artificial viscosity (used for numerical stability), which can lead to an incorrect formulation of the problem originally deemed inviscid.

Poisson's boundary conditions

The boundary condition they employ for self-gravity is periodic. They describe this as mimicking an infinite region of repeating boxes. Thus, when they model a cloud in the box using these boundary conditions, the solver for the gravity within this box, takes into account the gravitational effects of an infinite number of other spheres in boxes. To minimize the effect of these other spheres, they suggest making the distance between the center-to-center images d=4R. By increasing the d=8R, results in less gravitational pull by the mirrored clouds that hastens the collapse by only 1%.

Fedderath et al. (‘10) - Sink Particles

​Modeling Collapse and Accretion in Turbulent Gas Clouds: Implementation and Comparison of Sink Particles in AMR and SPH

Background

The numerical difficulty with modeling the collapse of a clump, while keeping track of the entire cloud, is given by the fact that the free-fall time, Tff, where

where

Tff = (3\pi / 32G\rho)^{1/2} $

decreases with increasing density and so resolution of the subgrids is demanded over many dynamical time scales. The two methods thus far designed to deal with this matter are 'Jeans heating' and 'sink particles'. Sink particles are a more realistic methodology and was first developed for SPH by Bate et al. ('95). The algorithm was later adapted by Krumholz et al. ('04) for Eulerian AMR. This paper describes a more rigorous series of checks for sink formation.

Sink Implentation

Sink particles enable the star formation rate/star formation efficiency, and mass distribution, to be addressed in a robust and quantitative way.

Sink algorithms originated from the notion of a density threshold. In earlier work, such a threshold was defined, and once surpassed, a sink particle was placed in the grid.

The present work, however, has added in addition to this criterion, a series of checks to insure that sinks are formed only in gravitationally bound and collapsing structures. These are listed as follows:

• Converging flow
• Bound system
• Jean's unstable
• Gravitational potential minima

Else, under conditions such as shear, a sink may erroneously form.

Tests

Tests included collapse of Bonner Ebert and singular isothermal spheres.

Self Similarity Solutions for Collapse

Shu (‘77)

​Self-similar collapse of isothermal spheres and star formation

Hunter (‘77)

​The collapse of unstable isothermal spheres

Larson & Penston et al. (‘04)

[​http://adsabs.harvard.edu/abs/1969MNRAS.144..425P Dynamics of self-gravitating gaseous spheres-III. Analytical results in the free-fall of isothermal case] [

Bonnor Ebert Spheres

Banerjee & Pudritz et al. (‘04)

​The formation and evolution of protostellar discs; three-dimensional adaptive mesh refinement hydrosimulations of collapsing, rotating Bonnor-Ebert spheres

Summary

3d hydro, AMR simulations of rotating Bonnor Ebert spheres with varying degrees of initial rotation. Final collapse structures depend on the degree of rotation, with the general trend being slower rotation leads to bars/filaments, faster rotation leads to rings. Radiative cooling is incorporated into the model and a more complicated EOS.

Methods

Both high and low mass BE spheres were studied. All runs had an m=2 density perturbation. The EOS was more sophisticated than earlier works in that it was a function of density and time.

Results

No fragmentation/non-central collapse in the isothermal regime.

Bars or rings form after the 'non-isothermal core' accretes surrounding gas and forms a disc.

Temperature shocks form after the isothermal phase; as the core begins to hit up and halt the collapse, cold material from the envelope falls in and forms a shock front of temperature discontinuity. The inner shock front sets the conditions that control subsequent disc structure and dynamics.

Foster and Chevalier (‘93)

​Gravitational Collapse of an Isothermal Sphere

Motivation

While the collapse problem of an isothermal sphere is well studied, there remains some conflict on the correct analytic solution for the collapse. Larson and Penston (1969) produced a solution for the collapse problem that was later criticized by Shu (1977) as being ad hoc in some of the mathematical formulations. Later numerical studies by Hunter (1977), however, showed that the collapse does follow the LP solution. Despite this, the Shu solution remains the prominent treatment of the problem. The authors here set out to check which solution indeed describes the collapse problem.

Summary

The Shu solution begins from core formation (born out of a singular isothermal sphere) and initiates collapse into the medium through an 'expansion wave'. In contrast, the LP solution follows the collapse of a uniformly dense, static sphere prior to and up to core formation. Hunter, however, found that the LP solution can in fact be extended past core formation. The Shu solution evolves to a solution similar to homologous inflow like the LP solution.

Foster and Chevalier ran simulations of a Bonner Ebert sphere with an outer density of 1/1000 the density at the outermost edge of the clump (my simulations used 1/100). The ambient therefore had a large sound speed throughout the constant pressure medium, that remained nearly constant throughout. Their results are robustly similar to the LP solution.

Stability Tests

The 'marginally stable sphere' had a xi=xiCrit=6.451. A density enhancement of 1% above equilibrium value was needed to bring out expected stability behavior:

• xi = 3, flow oscillated about v=0 for ~ 15 free fall times
• xi = 6.451, collapse happened in 5 ff times
• xi = 20, collapse happened in 6 ff times

Collapse Results

• Inflow is initially marked by velocity proportional to r for small r, peaking at xi = 2, and turning back to 0 velocity out toward edge of sphere.

• Simulation approaches LP solution in center region.

• Velocity smoothly transitions from homologous inflow to an inflow rate of ~ -3Cs after core formation.

• Inner region evolves fast, outer evolves slow.

• Core formation is not a subsonic adjustment of density with radius as Shu said. The inner regions collapse to a core supersonically.

• Varying the boundary conditions/initial conditions did not change the behavior from tending to the LP solution. Likewise, neither did changing the perturbation conditions. The mass accretion rate, however, if it became constant, was proportional to the magnitude of the perturbation as Shu found.

• Should compare anything to mass accretion rate?

Molecular Clouds

Turbulence