Sergey D. Ustyugov

COMPARING NUMERICAL METHODS FOR ISOTHERMAL MAGNETIZED SUPERSONIC TURBULENCE

Many astrophysical applications involve magnetized turbulent flows with shock waves. Ab initio star formation simulations require a robust representation of supersonic turbulence in molecular clouds on a wide range of scales imposing stringent demands on the quality of numerical algorithms. We employ simulations of supersonic super-Alfvenic turbulence decay as a benchmark test problem to assess and compare the performance of nine popular astrophysical MHD methods actively used to model star formation. The set of nine codes includes: ENZO, FLASH, KT-MHD, LL-MHD, PLUTO, PPML, RAMSES, STAGGER, and ZEUS. These applications employ a variety of numerical approaches, including both split and unsplit, finite difference and finite volume, divergence preserving and divergence cleaning, a variety of Riemann solvers, and a range of spatial reconstruction and time integration techniques. We present a comprehensive set of statistical measures designed to quantify the effects of numerical dissipation in these MHD solvers. We compare power spectra for basic fields to determine the effective spectral bandwidth of the methods and rank them based on their relative effective Reynolds numbers. We also compare numerical dissipation for solenoidal and dilatational velocity components to check for possible impacts of the numerics on small-scale density statistics. Finally, we discuss the convergence of various characteristics for the turbulence decay test and the impact of various components of numerical schemes on the accuracy of solutions. The nine codes gave qualitatively the same results, implying that they are all performing reasonably well and are useful for scientific applications. We show that the best performing codes employ a consistently high order of accuracy for spatial reconstruction of the evolved fields, transverse gradient interpolation, conservation law update step, and Lorentz force computation. The best results are achieved with divergence-free evolution of the magnetic field using the constrained transport method and using little to no explicit artificial viscosity. Codes that fall short in one or more of these areas are still useful, but they must compensate for higher numerical dissipation with higher numerical resolution. This paper is the largest, most comprehensive MHD code comparison on an application-like test problem to date. We hope this work will help developers improve their numerical algorithms while helping users to make informed choices about choosing optimal applications for their specific astrophysical problems.

THE TWO STATES OF STAR-FORMING CLOUDS

We examine the effects of self-gravity and magnetic fields on supersonic turbulence in isothermal molecular clouds with high-resolution simulations and adaptive mesh refinement. These simulations use large root grids (5123) to capture turbulence and four levels of refinement to follow the collapse to high densities, for an effective resolution of 81923. Three Mach 9 simulations are performed, two super-Alfv?nic and one trans-Alfv?nic. We find that gravity splits the clouds into two populations, one low-density turbulent state and one high-density collapsing state. The low-density state exhibits properties similar to non-self-gravitating in this regime, and we examine the effects of varied magnetic field strength on statistical properties: the density probability distribution function is approximately lognormal, the velocity power spectral slopes decrease with decreasing mean field strength, the alignment between velocity and magnetic field increases with the field, and the magnetic field probability distribution can be fitted to a stretched exponential. The high-density state is well characterized by self-similar spheres: the density probability distribution is a power law, collapse rate decreases with increasing mean field, density power spectra have positive slopes, P(?, k)k, thermal-to-magnetic pressure ratios are roughly unity for all mean field strengths, dynamic-to-magnetic pressure ratios are larger than unity for all mean field strengths, the magnetic field distribution follows a power-law distribution. The high Alfv?n Mach numbers in collapsing regions explain the recent observations of magnetic influence decreasing with density. We also find that the high-density state is typically found in filaments formed by converging flows, consistent with recent Herschel observations. Possible modifications to existing star formation theories are explored. The overall trans-Alfv?nic nature of star-forming clouds is discussed.

Simulating supersonic turbulence in magnetized molecular clouds

We present results of large-scale three-dimensional weakly magnetized supersonic turbulence simulations with an isothermal equation of state at grid resolutions up to 10243 cells with the Piecewise Parabolic Method on a Local Stencil. The turbulence is driven by a large-scale isotropic solenoidal force in a periodic computational domain and fully develops in a few flow crossing times. We then evolve the flow for a number of flow crossing times and analyze various statistical properties of the saturated turbulent state. We show that the energy transfer rate in the inertial range of scales is surprisingly close to a constant, indicating that Kolmogorovs phenomenology for incompressible turbulence can be extended to magnetized supersonic flows. We also discuss numerical dissipation effects and convergence of different turbulence diagnostics as grid resolution refines from 2563 to 10243 cells.

MHD Turbulence In Star‐Forming Clouds

Supersonic magneto‐hydrodynamic (MHD) turbulence in molecular clouds (MCs) plays an important role in the process of star formation. The effect of the turbulence on the cloud fragmentation process depends on the magnetic field strength. In this work we discuss the idea that the turbulence is super‐Alfvenic, at least with respect to the cloud mean magnetic field. We argue that MCs are likely to be born super‐Alfvenic. We then support this scenario based on a recent simulation of the large‐scale warm interstellar medium turbulence. Using small‐scale isothermal MHD turbulence simulation, we also show that MCs may remain super‐Alfvenic even with respect to their rms magnetic field strength, amplified by the turbulence. Finally, we briefly discuss the comparison with the observations, suggesting that super‐Alfvenic turbulence successfully reproduces the Zeeman measurements of the magnetic field strength in dense MC clouds.

Development of the geometric structure of the thermonuclear-deflagration front in type Ia supernovae

Three-dimensional hydrodynamical simulations of the development of a large-scale instability accompanying deflagration in the degenerate cores of rotating white dwarfs—progenitors of type-Ia supernovae—are presented. The numerical algorithm used is described in detail. An explicit, conservative, Godunov-type TVD difference scheme was employed for the computations. Large-scale convective processes are important as the deflagration front propagates. The supernova explosion is strongly nonspherically symmetric; a large-scale front structure emerges and propagates most rapidly along the rotational axis. The arrival of fresh thermonuclear fuel to the central region of the core can result in flares and the destruction of the core.

Realistic magnetohydrodynamical simulation of solar local supergranulation

Three-dimensional numerical simulations of solar surface magnetoconvection using realistic model physics are conducted. The thermal structure of convective motions into the upper radiative layers of the photosphere, the main scales of convective cells and the penetration depths of convection are investigated. We take part of the solar photosphere with a size of 60×60 Mm2 in the horizontal direction and of depth 20 Mm from the level of the visible solar surface. We use a realistic initial model of the sun and apply the equation of state and opacities of stellar matter. The equations of fully compressible radiation magnetohydrodynamics (MHD) with dynamical viscosity and gravity are solved. We apply (i) the conservative total variation diminishing (TVD) difference scheme for MHD, (ii) the diffusion approximation for radiative transfer and (iii) dynamical viscosity from subgrid-scale modeling. In simulation, we take a uniform two-dimensional grid in the horizontal plane and a nonuniform grid in the vertical direction with the number of cells being 600×600×204. We use 512 processors with distributed memory multiprocessors on the supercomputer MVS-100k at the Joint Computational Centre of the Russian Academy of Sciences.

The structure and statistics of interstellar turbulence

We explore the structure and statistics of multiphase, magnetized ISM turbulence in the local Milky Way by means of driven periodic box numerical MHD simulations. Using the higher order-accurate piecewise-parabolic method on a local stencil (PPML), we carry out a small parameter survey varying the mean magnetic field strength and density while fixing the rms velocity to observed values. We quantify numerous characteristics of the transient and steady-state turbulence, including its thermodynamics and phase structure, kinetic and magnetic energy power spectra, structure functions, and distribution functions of density, column density, pressure, and magnetic field strength. The simulations reproduce many observables of the local ISM, including molecular clouds, such as the ratio of turbulent to mean magnetic field at 100 pc scale, the mass and volume fractions of thermally stable HI, the lognormal distribution of column densities, the mass-weighted distribution of thermal pressure, and the linewidth-size relationship for molecular clouds. Our models predict the shape of magnetic field probability density functions (PDFs), which are strongly non-Gaussian, and the relative alignment of magnetic field and density structures. Finally, our models show how the observed low rates of star formation per free-fall time are controlled by the multiphase thermodynamics and large-scale turbulence.

Interstellar Turbulence and Star Formation

We provide a brief overview of recent advances and outstanding issues in simulations of interstellar turbulence, including isothermal models for interior structure of molecular clouds and larger-scale multiphase models designed to simulate the formation of molecular clouds. We show how self-organization in highly compressible magnetized turbulence in the multiphase ISM can be exploited in simple numerical models to generate realistic initial conditions for star formation.

Boundary conditions for simulations of the thermal outburst of a type Ia supernova

We present a technique to calculate the boundary conditions for simulations of the development of large-scale convective instability in the cores of rotating white-dwarf progenitors of type Ia supernovae. The hydrodynamical equations describing this situation are analyzed. We also study the impact of the boundary conditions on the development of the thermal outburst.

Piecewise Parabolic Method on a Local Stencil for Gas Dynamics and MHD

A numerical method for solving hyperbolic equations which is based on the piecewise parabolic approximation is presented. We use the property of the Riemann invariants to remain constant along the characteristics of the equations and utilize information from the previous time step in order to reconstruct a solution inside the difference cells. This approach allows us to exclude the interpolation procedure on the four point stencil used in the original piecewise parabolic method (PPM). It also provides the ability to exactly represent discontinuous solutions without adding excessive dissipation.