skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.


Title: Finite shock model of density in supersonic turbulence
ABSTRACT The probability distribution of density in isothermal, supersonic, turbulent gas is approximately lognormal. This behaviour can be traced back to the shock waves travelling through the medium, which randomly adjust the density by a random factor of the local sonic Mach number squared. Provided a certain parcel of gas experiences a large number of shocks, due to the central limit theorem, the resulting distribution for density is lognormal. We explore a model in which parcels of gas undergo finite number of shocks before relaxing to the ambient density, causing the distribution for density to deviate from a lognormal. We confront this model with numerical simulations with various rms Mach numbers ranging from subsonic as low as 0.1 to supersonic at 25. We find that the fits to the finite formula are an order of magnitude better than a lognormal. The model naturally extends even to subsonic flows, where no shocks exist.  more » « less
Award ID(s):
1616026
PAR ID:
10400978
Author(s) / Creator(s):
;
Publisher / Repository:
Oxford University Press
Date Published:
Journal Name:
Monthly Notices of the Royal Astronomical Society: Letters
Volume:
521
Issue:
1
ISSN:
1745-3925
Format(s):
Medium: X Size: p. L64-L69
Size(s):
p. L64-L69
Sponsoring Org:
National Science Foundation
More Like this
  1. null (Ed.)
    ABSTRACT We report three-dimensional hydrodynamical simulations of shocks ($${\cal M_{\rm shock}}\ge 4$$) interacting with fractal multicloud layers. The evolution of shock–multicloud systems consists of four stages: a shock-splitting phase in which reflected and refracted shocks are generated, a compression phase in which the forward shock compresses cloud material, an expansion phase triggered by internal heating and shock re-acceleration, and a mixing phase in which shear instabilities generate turbulence. We compare multicloud layers with narrow ($$\sigma _{\rho }=1.9\bar{\rho }$$) and wide ($$\sigma _{\rho }=5.9\bar{\rho }$$) lognormal density distributions characteristic of Mach ≈ 5 supersonic turbulence driven by solenoidal and compressive modes. Our simulations show that outflowing cloud material contains imprints of the density structure of their native environments. The dynamics and disruption of multicloud systems depend on the porosity and the number of cloudlets in the layers. ‘Solenoidal’ layers mix less, generate less turbulence, accelerate faster, and form a more coherent mixed-gas shell than the more porous ‘compressive’ layers. Similarly, multicloud systems with more cloudlets quench mixing via a shielding effect and enhance momentum transfer. Mass loading of diffuse mixed gas is efficient in all models, but direct dense gas entrainment is highly inefficient. Dense gas only survives in compressive clouds, but has low speeds. If normalized with respect to the shock-passage time, the evolution shows invariance for shock Mach numbers ≥10 and different cloud-generating seeds, and slightly weaker scaling for lower Mach numbers and thinner cloud layers. Multicloud systems also have better convergence properties than single-cloud systems, with a resolution of eight cells per cloud radius being sufficient to capture their overall dynamics. 
    more » « less
  2. Abstract Supersonic isothermal turbulence is a common process in astrophysical systems. In this work, we explore the energy in such systems. We show that the conserved energy is the sum of the kinetic energy (K) and Helmholtz free energy (F). We develop analytic predictions for the probability distributions,P(F) andP(K), as well as their nontrivial joint distribution,P(F,K). We verify these predictions with a suite of driven turbulence simulations, finding excellent agreement. The turbulence simulations were performed at Mach numbers ranging from 1 to 8, and three modes of driving: purely solenoidal, purely compressive, and mixed. We find thatP(F) is discontinuous atF= 0, with the discontinuity increasing with Mach number and compressive driving.P(K) resembles a lognormal with a negative skew. The joint distribution,P(F,K), shows a bimodal distribution, with gas either existing at highFand highKor at lowFand lowK. 
    more » « less
  3. ABSTRACT In star-forming clouds, high velocity flow gives rise to large fluctuations of density. In this work, we explore the correlation between velocity magnitude (speed) and density. We develop an analytic formula for the joint probability distribution function (PDF) of density and speed, and discuss its properties. In order to develop an accurate model for the joint PDF, we first develop improved models of the marginalized distributions of density and speed. We confront our results with a suite of 12 supersonic isothermal simulations with resolution of $1024^3$ cells in which the turbulence is driven by 3 different forcing modes (solenoidal, mixed, and compressive) and 4 rms Mach numbers (1, 2, 4, 8). We show, that for transsonic turbulence, density and speed are correlated to a considerable degree and the simple assumption of independence fails to accurately describe their statistics. In the supersonic regime, the correlations tend to weaken with growing Mach number. Our new model of the joint and marginalized PDFs are a factor of 3 better than uncorrelated, and provides insight into this important process. 
    more » « less
  4. Abstract Using a suite of 3D hydrodynamical simulations of star-forming molecular clouds, we investigate how the density probability distribution function (PDF) changes when including gravity, turbulence, magnetic fields, and protostellar outflows and heating. We find that the density PDF is not lognormal when outflows and self-gravity are considered. Self-gravity produces a power-law tail at high densities, and the inclusion of stellar feedback from protostellar outflows and heating produces significant time-varying deviations from a lognormal distribution at low densities. The simulation with outflows has an excess of diffuse gas compared to the simulations without outflows, exhibits an increased average sonic Mach number, and maintains a slower star formation rate (SFR) over the entire duration of the run. We study the mass transfer between the diffuse gas in the lognormal peak of the PDF, the collapsing gas in the power-law tail, and the stars. We find that the mass fraction in the power-law tail is constant, such that the stars form out of the power-law gas at the same rate at which the gas from the lognormal part replenishes the power law. We find that turbulence does not provide significant support in the dense gas associated with the power-law tail. When including outflows and magnetic fields in addition to driven turbulence, the rate of mass transfer from the lognormal to the power law, and then to the stars, becomes significantly slower, resulting in slower SFRs and longer depletion times. 
    more » « less
  5. Abstract We use a suite of 3D simulations of star-forming molecular clouds, with and without stellar feedback, magnetic fields, and driven turbulence, to study the compression and expansion rates of the gas as functions of density. We show that, around the mean density, supersonic turbulence promotes rough equilibrium between the amounts of compressing and expanding gas, consistent with continuous gas cycling between high- and low-density states. We find that the inclusion of protostellar jets produces rapidly expanding and compressing low-density gas. We find that the gas mass flux peaks at the transition between the lognormal and power-law forms of the density probability distribution function (PDF). This is consistent with the transition density tracking the post-shock density, which promotes an enhancement of mass at this density (i.e., shock compression and filament formation). At high densities, the gas dynamics are dominated by self-gravity: the compression rate in all of our runs matches the rate of the run with only gravity, suggesting that processes other than self-gravity have little effect at these densities. The net gas mass flux becomes constant at a density below the sink formation threshold, where it equals the star formation rate. The density at which the net gas mass flux equals the star formation rate is one order of magnitude lower than our sink threshold density, corresponds to the formation of the second power-law tail in the density PDF, and sets the overall star formation rates of these simulations. 
    more » « less