Liubin Pan

TURBULENCE-INDUCED RELATIVE VELOCITY OF DUST PARTICLES. I. IDENTICAL PARTICLES

We study the relative velocity of inertial particles suspended in turbulent flows and discuss implications for dust particle collisions in protoplanetary disks. We simulate a weakly compressible turbulent flow, evolving 14 particle species with friction timescale, τp, covering the entire range of scales in the flow. The particle Stokes numbers, St, measuring the ratio of τp to the Kolmogorov timescale, are in the range 0.1 St 800. Using simulation results, we show that the model by Pan & Padoan gives satisfactory predictions for the rms relative velocity between identical particles. The probability distribution function (PDF) of the relative velocity is found to be highly non-Gaussian. The PDF tails are well described by a 4/3 stretched exponential function for particles with τp 1-2 T L, where T L is the Lagrangian correlation timescale, consistent with a prediction based on PP10. The PDF approaches Gaussian only for very large particles with τp 54 T L. We split particle pairs at given distances into two types with low and high relative speeds, referred to as continuous and caustic types, respectively, and compute their contributions to the collision kernel. Although amplified by the effect of clustering, the continuous contribution vanishes in the limit of infinitesimal particle distance, where the caustic contribution dominates. The caustic kernel per unit cross section rises rapidly as St increases toward 1, reaches a maximum at τp 2 T L, and decreases as for τp T L.

MIXING OF CLUMPY SUPERNOVA EJECTA INTO MOLECULAR CLOUDS

Several lines of evidence, from isotopic analyses of meteorites to studies of the Suns elemental and isotopic composition, indicate that the solar system was contaminated early in its evolution by ejecta from a nearby supernova. Previous models have invoked supernova material being injected into an extant protoplanetary disk, or isotropically expanding ejecta sweeping over a distant (>10 pc) cloud core, simultaneously enriching it and triggering its collapse. Here, we consider a new astrophysical setting: the injection of clumpy supernova ejecta, as observed in the Cassiopeia A supernova remnant, into the molecular gas at the periphery of an H II region created by the supernovas progenitor star. To track these interactions, we have conducted a suite of high-resolution (15003 effective) three-dimensional numerical hydrodynamic simulations that follow the evolution of individual clumps as they move into molecular gas. Even at these high resolutions, our simulations do not quite achieve numerical convergence, due to the challenge of properly resolving the small-scale mixing of ejecta and molecular gas, although they do allow some robust conclusions to be drawn. Isotropically exploding ejecta do not penetrate into the molecular cloud or mix with it, but, if cooling is properly accounted for, clumpy ejecta penetrate to distances ~1018 cm and mix effectively with large regions of star-forming molecular gas. In fact, the ~2 M ☉ of high-metallicity ejecta from a single core-collapse supernova is likely to mix with ~2 × 104 M ☉ of molecular gas material as it is collapsing. Thus, all stars forming late (5 Myr) in the evolution of an H II region may be contaminated by supernova ejecta at the level ~10–4. This level of contamination is consistent with the abundances of short-lived radionuclides and possibly some stable isotopic shifts in the early solar system and is potentially consistent with the observed variability in stellar elemental abundances. Supernova contamination of forming planetary systems may be a common, universal process.

Mixing in supersonic turbulence

In many astrophysical environments, mixing of heavy elements occurs in the presence of a supersonic turbulent velocity field. Here, we carry out the first systematic numerical study of such passive scalar mixing in isothermal supersonic turbulence. Our simulations show that the ratio of the scalar mixing timescale, ?c, to the flow dynamical time, ?dyn (defined as the flow driving scale divided by the rms velocity), increases with the Mach number, M, for M 3, and becomes essentially constant for M 3. This trend suggests that compressible modes are less efficient in enhancing mixing than solenoidal modes. However, since the majority of kinetic energy is contained in solenoidal modes at all Mach numbers, the overall change in ?c/?dyn is less than 20% over the range 1 M 6. At all Mach numbers, if pollutants are injected at around the flow driving scale, ?c is close to ?dyn. This suggests that scalar mixing is driven by a cascade process similar to that of the velocity field. The dependence of ?c on the length scale at which pollutants are injected into flow is also consistent with this cascade picture. Similar behavior is found for the variance decay timescales for scalars without continuing sources. Extension of the scalar cascade picture to the supersonic regime predicts a relation between the scaling exponents of the velocity and the scalar structure functions, with the scalar structure function becoming flatter as the velocity scaling steepens with Mach number. Our measurements of the volume-weighted velocity and scalar structure functions confirm this relation for M 2, but show discrepancies at M 3, which arise probably because strong expansions and compressions tend to make scalar structure functions steeper.

SUPERNOVA DRIVING. II. COMPRESSIVE RATIO IN MOLECULAR-CLOUD TURBULENCE

The compressibility of molecular cloud (MC) turbulence plays a crucial role in star formation models, because it controls the amplitude and distribution of density fluctuations. The relation between the compressive ratio (the ratio of powers in compressive and solenoidal motions) and the statistics of turbulence has been previously studied systematically only in idealized simulations with random external forces. In this work, we analyze a simulation of large-scale turbulence (250 pc) driven by supernova (SN) explosions that has been shown to yield realistic MC properties. We demonstrate that SN driving results in MC turbulence with a broad lognormal distribution of the compressive ratio, with a mean value

MIXING IN MAGNETIZED TURBULENT MEDIA

\approx 0.3

IDENTIFICATION OF A FUNDAMENTAL TRANSITION IN A TURBULENTLY SUPPORTED INTERSTELLAR MEDIUM

, lower than the equilibrium value of

Passive scalar structures in supersonic turbulence.

\approx 0.5

TURBULENCE-INDUCED RELATIVE VELOCITY OF DUST PARTICLES. IV. THE COLLISION KERNEL

found in the inertial range of isothermal simulations with random solenoidal driving. We also find that the compressibility of the turbulence is not noticeably affected by gravity, nor are the mean cloud radial (expansion or contraction) and solid-body rotation velocities. Furthermore, the clouds follow a general relation between the rms density and the rms Mach number similar to that of supersonic isothermal turbulence, though with a large scatter, and their average gas density PDF is described well by a lognormal distribution, with the addition of a high-density power-law tail when self-gravity is included.

SUPERNOVA DRIVING. III. SYNTHETIC MOLECULAR CLOUD OBSERVATIONS

Turbulent motions are essential to the mixing of entrained fluids and are also capable of amplifying weak initial magnetic fields by small-scale dynamo action. Here we perform a systematic study of turbulent mixing in magnetized media, using three-dimensional magnetohydrodynamic simulations that include a scalar concentration field. We focus on how mixing depends on the magnetic Prandtl number, Pm, from 1 to 4 and the Mach number, from 0.3 to 2.4. For all subsonic flows, we find that the velocity power spectrum has a k –5/3 slope in the early kinematic phase, but steepens due to magnetic back reactions as the field saturates. The scalar power spectrum, on the other hand, flattens compared to k –5/3 at late times, consistent with the Obukohov-Corrsin picture of mixing as a cascade process. At higher Mach numbers, the velocity power spectrum also steepens due to the presence of shocks, and the scalar power spectrum again flattens accordingly. Scalar structures are more intermittent than velocity structures in subsonic turbulence, whereas for supersonic turbulence, velocity structures appear more intermittent than the scalars only in the kinematic phase. Independent of the Mach number of the flow, scalar structures are arranged in sheets in both the kinematic and saturated phases of the magnetic field evolution. For subsonic turbulence, scalar dissipation is hindered in the strong magnetic field regions, probably due to Lorentz forces suppressing the buildup of scalar gradients, whereas for supersonic turbulence, scalar dissipation increases monotonically with increasing magnetic field strength. At all Mach numbers, mixing is significantly slowed by the presence of dynamically important small-scale magnetic fields, implying that mixing in the interstellar medium and in galaxy clusters is less efficient than modeled in hydrodynamic simulations.

The pollution of pristine material in compressible turbulence

The interstellar medium (ISM) in star-forming galaxies is a multiphase gas in which turbulent support is at least as important as thermal pressure. Sustaining this configuration requires continuous radiative cooling, such that the overall average cooling rate matches the decay rate of turbulent energy into the medium. Here we carry out a set of numerical simulations of a stratified, turbulently stirred, radiatively cooled medium, which uncover a fundamental transition at a critical one-dimensional turbulent velocity of ≈35 km s–1. At turbulent velocities below ≈35 km s–1, corresponding to temperatures below 105.5 K, the medium is stable, as the time for gas to cool is roughly constant as a function of temperature. On the other hand, at turbulent velocities above the critical value, the gas is shocked into an unstable regime in which the cooling time increases strongly with temperature, meaning that a substantial fraction of the ISM is unable to cool on a turbulent dissipation timescale. This naturally leads to runaway heating and ejection of gas from any stratified medium with a 1D turbulent velocity above ≈35 km s–1, a result that has implications for galaxy evolution at all redshifts.